%------------------------------------------------------------------------------
% File : iProver---3.9.4
% Problem : SEU344+1 : TPTP v9.3.1. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_iprover 300 /export/starexec/sandbox2/benchmark/theBenchmark.p THM
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Fri Sep 25 02:56:59 PM UTC 2026
% Result : Theorem 1.10s 1.09s
% Output : CNFRefutation 1.10s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 8
% Syntax : Number of formulae : 70 ( 17 unt; 0 def)
% Number of atoms : 211 ( 43 equ)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 235 ( 94 ~; 105 |; 19 &)
% ( 5 <=>; 11 =>; 0 <=; 1 <~>)
% Maximal formula depth : 13 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of types : 1 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 10 ( 8 usr; 1 prp; 0-3 aty)
% Number of functors : 9 ( 9 usr; 3 con; 0-3 aty)
% Number of variables : 87 ( 0 sgn 78 !; 9 ?; 28 :)
% Comments :
%------------------------------------------------------------------------------
fof(f10,axiom,
! [X0] :
( ( join_semilatt_str(X0)
& ~ empty_carrier(X0) )
=> ! [X1] :
( element(X1,the_carrier(X0))
=> ! [X2] :
( element(X2,the_carrier(X0))
=> ( below(X0,X1,X2)
<=> join(X0,X1,X2) = X2 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d3_lattices) ).
fof(f15,axiom,
! [X0] :
( latt_str(boole_lattice(X0))
& strict_latt_str(boole_lattice(X0)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_k1_lattice3) ).
fof(f30,axiom,
! [X0] :
( latt_str(X0)
=> ( join_semilatt_str(X0)
& meet_semilatt_str(X0) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',dt_l3_lattices) ).
fof(f44,axiom,
! [X0] :
( strict_latt_str(boole_lattice(X0))
& ~ empty_carrier(boole_lattice(X0)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fc1_lattice3) ).
fof(f68,axiom,
! [X0,X1] :
( subset(X0,X1)
=> set_union2(X0,X1) = X1 ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t12_xboole_1) ).
fof(f70,axiom,
! [X0,X1] :
( element(X1,the_carrier(boole_lattice(X0)))
=> ! [X2] :
( element(X2,the_carrier(boole_lattice(X0)))
=> ( meet(boole_lattice(X0),X1,X2) = set_intersection2(X1,X2)
& join(boole_lattice(X0),X1,X2) = set_union2(X1,X2) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_lattice3) ).
fof(f73,conjecture,
! [X0,X1] :
( element(X1,the_carrier(boole_lattice(X0)))
=> ! [X2] :
( element(X2,the_carrier(boole_lattice(X0)))
=> ( below(boole_lattice(X0),X1,X2)
<=> subset(X1,X2) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t2_lattice3) ).
fof(f74,negated_conjecture,
~ ! [X0,X1] :
( element(X1,the_carrier(boole_lattice(X0)))
=> ! [X2] :
( element(X2,the_carrier(boole_lattice(X0)))
=> ( below(boole_lattice(X0),X1,X2)
<=> subset(X1,X2) ) ) ),
inference(negated_conjecture,[status(cth)],[f73]) ).
fof(f81,axiom,
! [X0,X1] : subset(X0,set_union2(X0,X1)),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t7_xboole_1) ).
fof(f86,plain,
? [X0,X1] :
( element(X1,the_carrier(boole_lattice(X0)))
& ? [X2] :
( element(X2,the_carrier(boole_lattice(X0)))
& ( below(boole_lattice(X0),X1,X2)
<~> subset(X1,X2) ) ) ),
inference(ennf_transformation,[],[f74]) ).
fof(f87,plain,
! [X0] :
( ~ join_semilatt_str(X0)
| empty_carrier(X0)
| ! [X1] :
( ~ element(X1,the_carrier(X0))
| ! [X2] :
( ~ element(X2,the_carrier(X0))
| ( below(X0,X1,X2)
<=> join(X0,X1,X2) = X2 ) ) ) ),
inference(ennf_transformation,[],[f10]) ).
fof(f88,plain,
! [X0] :
( ~ join_semilatt_str(X0)
| empty_carrier(X0)
| ! [X1] :
( ~ element(X1,the_carrier(X0))
| ! [X2] :
( ~ element(X2,the_carrier(X0))
| ( below(X0,X1,X2)
<=> join(X0,X1,X2) = X2 ) ) ) ),
inference(flattening,[],[f87]) ).
fof(f89,plain,
! [X0,X1] :
( ~ subset(X0,X1)
| set_union2(X0,X1) = X1 ),
inference(ennf_transformation,[],[f68]) ).
fof(f90,plain,
! [X0,X1] :
( ~ element(X1,the_carrier(boole_lattice(X0)))
| ! [X2] :
( ~ element(X2,the_carrier(boole_lattice(X0)))
| ( meet(boole_lattice(X0),X1,X2) = set_intersection2(X1,X2)
& join(boole_lattice(X0),X1,X2) = set_union2(X1,X2) ) ) ),
inference(ennf_transformation,[],[f70]) ).
fof(f92,plain,
! [X0] :
( ~ latt_str(X0)
| ( join_semilatt_str(X0)
& meet_semilatt_str(X0) ) ),
inference(ennf_transformation,[],[f30]) ).
fof(f134,plain,
? [X0,X1] :
( element(X1,the_carrier(boole_lattice(X0)))
& ? [X2] :
( element(X2,the_carrier(boole_lattice(X0)))
& ( below(boole_lattice(X0),X1,X2)
| subset(X1,X2) )
& ( ~ below(boole_lattice(X0),X1,X2)
| ~ subset(X1,X2) ) ) ),
inference(nnf_transformation,[],[f86]) ).
fof(f135,plain,
? [X0,X1] :
( element(X1,the_carrier(boole_lattice(X0)))
& ? [X2] :
( element(X2,the_carrier(boole_lattice(X0)))
& ( below(boole_lattice(X0),X1,X2)
| subset(X1,X2) )
& ( ~ below(boole_lattice(X0),X1,X2)
| ~ subset(X1,X2) ) ) ),
inference(flattening,[],[f134]) ).
fof(f136,plain,
( element(sK1,the_carrier(boole_lattice(sK0)))
& element(sK2,the_carrier(boole_lattice(sK0)))
& ( below(boole_lattice(sK0),sK1,sK2)
| subset(sK1,sK2) )
& ( ~ below(boole_lattice(sK0),sK1,sK2)
| ~ subset(sK1,sK2) ) ),
inference(skolemize,[status(esa),new_symbols(skolem,[sK0,sK1,sK2]),skolemize(X0,sK0),skolemize(X1,sK1),skolemize(X2,sK2)],[f135]) ).
fof(f138,plain,
! [X0] :
( ~ join_semilatt_str(X0)
| empty_carrier(X0)
| ! [X1] :
( ~ element(X1,the_carrier(X0))
| ! [X2] :
( ~ element(X2,the_carrier(X0))
| ( ( ~ below(X0,X1,X2)
| join(X0,X1,X2) = X2 )
& ( join(X0,X1,X2) != X2
| below(X0,X1,X2) ) ) ) ) ),
inference(nnf_transformation,[],[f88]) ).
fof(f155,plain,
element(sK1,the_carrier(boole_lattice(sK0))),
inference(cnf_transformation,[],[f136]) ).
fof(f156,plain,
element(sK2,the_carrier(boole_lattice(sK0))),
inference(cnf_transformation,[],[f136]) ).
fof(f157,plain,
( below(boole_lattice(sK0),sK1,sK2)
| subset(sK1,sK2) ),
inference(cnf_transformation,[],[f136]) ).
fof(f158,plain,
( ~ below(boole_lattice(sK0),sK1,sK2)
| ~ subset(sK1,sK2) ),
inference(cnf_transformation,[],[f136]) ).
fof(f160,plain,
! [X2,X0,X1] :
( ~ join_semilatt_str(X0)
| empty_carrier(X0)
| ~ element(X1,the_carrier(X0))
| ~ element(X2,the_carrier(X0))
| ~ below(X0,X1,X2)
| join(X0,X1,X2) = X2 ),
inference(cnf_transformation,[],[f138]) ).
fof(f161,plain,
! [X2,X0,X1] :
( ~ join_semilatt_str(X0)
| empty_carrier(X0)
| ~ element(X1,the_carrier(X0))
| ~ element(X2,the_carrier(X0))
| join(X0,X1,X2) != X2
| below(X0,X1,X2) ),
inference(cnf_transformation,[],[f138]) ).
fof(f162,plain,
! [X0,X1] : subset(X0,set_union2(X0,X1)),
inference(cnf_transformation,[],[f81]) ).
fof(f165,plain,
! [X0,X1] :
( ~ subset(X0,X1)
| set_union2(X0,X1) = X1 ),
inference(cnf_transformation,[],[f89]) ).
fof(f168,plain,
! [X2,X0,X1] :
( ~ element(X1,the_carrier(boole_lattice(X0)))
| ~ element(X2,the_carrier(boole_lattice(X0)))
| join(boole_lattice(X0),X1,X2) = set_union2(X1,X2) ),
inference(cnf_transformation,[],[f90]) ).
fof(f170,plain,
! [X0] : ~ empty_carrier(boole_lattice(X0)),
inference(cnf_transformation,[],[f44]) ).
fof(f171,plain,
! [X0] : latt_str(boole_lattice(X0)),
inference(cnf_transformation,[],[f15]) ).
fof(f175,plain,
! [X0] :
( ~ latt_str(X0)
| join_semilatt_str(X0) ),
inference(cnf_transformation,[],[f92]) ).
tcf(c_49,negated_conjecture,
( ~ subset(sK1,sK2)
| ~ below(boole_lattice(sK0),sK1,sK2) ),
inference(cnf_transformation,[],[f158]) ).
tcf(c_50,negated_conjecture,
( subset(sK1,sK2)
| below(boole_lattice(sK0),sK1,sK2) ),
inference(cnf_transformation,[],[f157]) ).
tcf(c_51,negated_conjecture,
element(sK2,the_carrier(boole_lattice(sK0))),
inference(cnf_transformation,[],[f156]) ).
tcf(c_52,negated_conjecture,
element(sK1,the_carrier(boole_lattice(sK0))),
inference(cnf_transformation,[],[f155]) ).
tcf(c_54,plain,
! [X0: $i,X1: $i,X2: $i] :
( empty_carrier(X0)
| below(X0,X1,X2)
| ~ join_semilatt_str(X0)
| ~ element(X2,the_carrier(X0))
| ~ element(X1,the_carrier(X0))
| ( join(X0,X1,X2) != X2 ) ),
inference(cnf_transformation,[],[f161]) ).
tcf(c_55,plain,
! [X0: $i,X1: $i,X2: $i] :
( empty_carrier(X0)
| ( join(X0,X1,X2) = X2 )
| ~ join_semilatt_str(X0)
| ~ element(X2,the_carrier(X0))
| ~ element(X1,the_carrier(X0))
| ~ below(X0,X1,X2) ),
inference(cnf_transformation,[],[f160]) ).
tcf(c_56,plain,
! [X0: $i,X1: $i] : subset(X0,set_union2(X0,X1)),
inference(cnf_transformation,[],[f162]) ).
tcf(c_59,plain,
! [X0: $i,X1: $i] :
( ( set_union2(X0,X1) = X1 )
| ~ subset(X0,X1) ),
inference(cnf_transformation,[],[f165]) ).
tcf(c_61,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( join(boole_lattice(X1),X0,X2) = set_union2(X0,X2) )
| ~ element(X2,the_carrier(boole_lattice(X1)))
| ~ element(X0,the_carrier(boole_lattice(X1))) ),
inference(cnf_transformation,[],[f168]) ).
tcf(c_63,plain,
! [X0: $i] : ~ empty_carrier(boole_lattice(X0)),
inference(cnf_transformation,[],[f170]) ).
tcf(c_66,plain,
! [X0: $i] : latt_str(boole_lattice(X0)),
inference(cnf_transformation,[],[f171]) ).
tcf(c_70,plain,
! [X0: $i] :
( join_semilatt_str(X0)
| ~ latt_str(X0) ),
inference(cnf_transformation,[],[f175]) ).
tcf(c_184,plain,
( ~ below(boole_lattice(sK0),sK1,sK2)
| ~ subset(sK1,sK2) ),
inference(prop_impl_just,[status(thm)],[c_49]) ).
tcf(c_185,plain,
( ~ subset(sK1,sK2)
| ~ below(boole_lattice(sK0),sK1,sK2) ),
inference(renaming,[status(thm)],[c_184]) ).
tcf(c_186,plain,
( below(boole_lattice(sK0),sK1,sK2)
| subset(sK1,sK2) ),
inference(prop_impl_just,[status(thm)],[c_50]) ).
tcf(c_187,plain,
( subset(sK1,sK2)
| below(boole_lattice(sK0),sK1,sK2) ),
inference(renaming,[status(thm)],[c_186]) ).
tcf(c_204,plain,
! [X0: $i] :
( ~ latt_str(X0)
| join_semilatt_str(X0) ),
inference(prop_impl_just,[status(thm)],[c_70]) ).
tcf(c_205,plain,
! [X0: $i] :
( join_semilatt_str(X0)
| ~ latt_str(X0) ),
inference(renaming,[status(thm)],[c_204]) ).
tcf(c_1280,plain,
! [X0: $i,X1: $i,X2: $i] :
( empty_carrier(X0)
| subset(sK1,sK2)
| ( join(X0,X1,X2) = X2 )
| ~ join_semilatt_str(X0)
| ~ element(X2,the_carrier(X0))
| ~ element(X1,the_carrier(X0))
| ( X2 != sK2 )
| ( X1 != sK1 )
| ( boole_lattice(sK0) != X0 ) ),
inference(resolution_lifted,[status(thm)],[c_187,c_55]) ).
tcf(c_1281,plain,
( empty_carrier(boole_lattice(sK0))
| subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) = sK2 )
| ~ join_semilatt_str(boole_lattice(sK0))
| ~ element(sK2,the_carrier(boole_lattice(sK0)))
| ~ element(sK1,the_carrier(boole_lattice(sK0))) ),
inference(unflattening,[status(thm)],[c_1280]) ).
tcf(c_1282,plain,
( empty_carrier(boole_lattice(sK0))
| subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) = sK2 )
| ~ join_semilatt_str(boole_lattice(sK0)) ),
inference(global_subsumption_just,[status(thm)],[c_1281,c_52,c_51,c_1281]) ).
tcf(c_1292,plain,
( subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) = sK2 )
| ~ join_semilatt_str(boole_lattice(sK0)) ),
inference(forward_subsumption_resolution,[status(thm)],[c_1282,c_63]) ).
tcf(c_1296,plain,
! [X0: $i,X1: $i,X2: $i] :
( empty_carrier(X0)
| ~ join_semilatt_str(X0)
| ~ subset(sK1,sK2)
| ~ element(X2,the_carrier(X0))
| ~ element(X1,the_carrier(X0))
| ( X2 != sK2 )
| ( X1 != sK1 )
| ( boole_lattice(sK0) != X0 )
| ( join(X0,X1,X2) != X2 ) ),
inference(resolution_lifted,[status(thm)],[c_185,c_54]) ).
tcf(c_1297,plain,
( empty_carrier(boole_lattice(sK0))
| ~ join_semilatt_str(boole_lattice(sK0))
| ~ subset(sK1,sK2)
| ~ element(sK2,the_carrier(boole_lattice(sK0)))
| ~ element(sK1,the_carrier(boole_lattice(sK0)))
| ( join(boole_lattice(sK0),sK1,sK2) != sK2 ) ),
inference(unflattening,[status(thm)],[c_1296]) ).
tcf(c_1298,plain,
( empty_carrier(boole_lattice(sK0))
| ~ join_semilatt_str(boole_lattice(sK0))
| ~ subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) != sK2 ) ),
inference(global_subsumption_just,[status(thm)],[c_1297,c_52,c_51,c_1297]) ).
tcf(c_1308,plain,
( ~ join_semilatt_str(boole_lattice(sK0))
| ~ subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) != sK2 ) ),
inference(forward_subsumption_resolution,[status(thm)],[c_1298,c_63]) ).
tcf(c_1621,plain,
! [X0: $i,X1: $i] :
( join_semilatt_str(X1)
| ( boole_lattice(X0) != X1 ) ),
inference(resolution_lifted,[status(thm)],[c_66,c_205]) ).
tcf(c_1622,plain,
! [X0: $i] : join_semilatt_str(boole_lattice(X0)),
inference(unflattening,[status(thm)],[c_1621]) ).
tcf(c_1770,plain,
( subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) = sK2 ) ),
inference(backward_subsumption_resolution,[status(thm)],[c_1292,c_1622]) ).
tcf(c_1771,plain,
( ~ subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) != sK2 ) ),
inference(backward_subsumption_resolution,[status(thm)],[c_1308,c_1622]) ).
tcf(c_2686,plain,
( ( join(boole_lattice(sK0),sK1,sK2) = sK2 )
| subset(sK1,sK2) ),
inference(prop_impl_just,[status(thm)],[c_1770]) ).
tcf(c_2687,plain,
( subset(sK1,sK2)
| ( join(boole_lattice(sK0),sK1,sK2) = sK2 ) ),
inference(renaming,[status(thm)],[c_2686]) ).
tcf(c_7064,plain,
! [X0: $i] :
( ( join(boole_lattice(sK0),sK1,X0) = set_union2(sK1,X0) )
| ~ element(X0,the_carrier(boole_lattice(sK0))) ),
inference(superposition,[status(thm)],[c_52,c_61]) ).
tcf(c_7177,plain,
join(boole_lattice(sK0),sK1,sK2) = set_union2(sK1,sK2),
inference(superposition,[status(thm)],[c_51,c_7064]) ).
tcf(c_8472,plain,
( subset(sK1,sK2)
| ( set_union2(sK1,sK2) = sK2 ) ),
inference(light_normalisation,[status(thm)],[c_2687,c_7177]) ).
tcf(c_8475,plain,
set_union2(sK1,sK2) = sK2,
inference(forward_subsumption_resolution,[status(thm)],[c_8472,c_59]) ).
tcf(c_8485,plain,
join(boole_lattice(sK0),sK1,sK2) = sK2,
inference(demodulation,[status(thm)],[c_7177,c_8475]) ).
tcf(c_8494,plain,
subset(sK1,sK2),
inference(superposition,[status(thm)],[c_8475,c_56]) ).
tcf(c_8506,plain,
$false,
inference(prop_impl_just,[status(thm)],[c_8494,c_8485,c_1771]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.01 % Problem : SEU344+1 : TPTP v9.3.1. Released v3.3.0.
% 0.00/0.02 % Command : run_iprover 300 /export/starexec/sandbox2/benchmark/theBenchmark.p THM
% 0.03/0.30 % Computer : n012.cluster.edu
% 0.03/0.30 % Model : x86_64 x86_64
% 0.03/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.03/0.30 % Memory : 8046.5625MB
% 0.03/0.30 % OS : Linux 6.8.0-71-generic
% 0.03/0.30 % CPULimit : 300
% 0.03/0.30 % WCLimit : 300
% 0.03/0.30 % DateTime : Thu Sep 24 13:46:20 UTC 2026
% 0.03/0.30 % CPUTime :
% 0.03/0.30 Running run_iprover 300 /export/starexec/sandbox2/benchmark/theBenchmark.p THM
% 0.03/0.33 Running first-order theorem proving
% 0.03/0.33 Running: /export/starexec/sandbox2/solver/bin/iproveropt-multi-core.sh -d -n -l tptp -s fof_schedule -t 300 /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.03/0.34
% 0.03/0.34 % ======== iProver multi-core TPTP/SMT =========
% 0.03/0.34
% 0.03/0.34 % Detected problem language: tptp
% 0.08/0.35 % Proving...
% 1.10/1.09 % SZS status Started for theBenchmark.p
% 1.10/1.09 % SZS status Theorem for theBenchmark.p
% 1.10/1.09
% 1.10/1.09 %---------------- iProver v3.9.4 (pre CASC 2026/SMT-COMP 2026) ----------------%
% 1.10/1.09
% 1.10/1.09 % ------ iProver source info
% 1.10/1.09
% 1.10/1.09 % git: date: 2026-07-19 20:42:38 +0200
% 1.10/1.09 % git: sha1: 804e7d636a263075307957e923b7a22a4035de61
% 1.10/1.09 % git: non_committed_changes: false
% 1.10/1.09
% 1.10/1.09 % ------ Parsing...
% 1.10/1.09 % ------ Clausification by vclausify_rel & Parsing by iProver...%
% 1.10/1.09
% 1.10/1.09 % ------ Preprocessing... sup_sim: 0 sf_s rm: 1 0s sf_e pe_s pe:1:0s pe:2:0s pe_e sup_sim: 0 sf_s rm: 3 0s sf_e pe_s pe_e %
% 1.10/1.09
% 1.10/1.09 % ------ Preprocessing... gs_s sp: 0 0s gs_e snvd_s sp: 0 0s snvd_e %
% 1.10/1.09
% 1.10/1.09 % ------ Preprocessing... sf_s rm: 1 0s sf_e sf_s rm: 0 0s sf_e
% 1.10/1.09 % ------ Proving...
% 1.10/1.09 % ------ Problem Properties
% 1.10/1.09
% 1.10/1.09 %
% 1.10/1.09 % clauses 79
% 1.10/1.09 % conjectures 2
% 1.10/1.09 % EPR 27
% 1.10/1.09 % Horn 69
% 1.10/1.09 % unary 33
% 1.10/1.09 % binary 24
% 1.10/1.09 % lits 188
% 1.10/1.09 % lits eq 22
% 1.10/1.09 % fd_pure 0
% 1.10/1.09 % fd_pseudo 0
% 1.10/1.09 % fd_cond 1
% 1.10/1.09 % fd_pseudo_cond 4
% 1.10/1.09 % AC symbols 0
% 1.10/1.09
% 1.10/1.09 % ------ Schedule dynamic 5 is on
% 1.10/1.09
% 1.10/1.09 % ------ Input Options "--resolution_flag false --inst_lit_sel_side none" Time Limit: 10.
% 1.10/1.09
% 1.10/1.09
% 1.10/1.09 % ------
% 1.10/1.09 % Current options:
% 1.10/1.09 % ------
% 1.10/1.09
% 1.10/1.09
% 1.10/1.09 %
% 1.10/1.09
% 1.10/1.09 % ------ Proving...
% 1.10/1.09 %
% 1.10/1.09
% 1.10/1.09 % SZS status Theorem for theBenchmark.p
% 1.10/1.09
% 1.10/1.09 % SZS output start CNFRefutation for theBenchmark.p
% See solution above
% 1.10/1.09
% 1.10/1.09
%------------------------------------------------------------------------------