TSTP Solution File: SEU106+1 by iProver---3.8

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : iProver---3.8
% Problem  : SEU106+1 : TPTP v8.1.2. Released v3.2.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : run_iprover %s %d THM

% Computer : n021.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 : Thu Aug 31 17:03:44 EDT 2023

% Result   : Theorem 3.72s 1.13s
% Output   : CNFRefutation 3.72s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   18
%            Number of leaves      :   10
% Syntax   : Number of formulae    :   66 (  23 unt;   0 def)
%            Number of atoms       :  197 (  17 equ)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives :  223 (  92   ~;  79   |;  37   &)
%                                         (   3 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   4 avg)
%            Maximal term depth    :    7 (   1 avg)
%            Number of predicates  :    6 (   4 usr;   1 prp; 0-2 aty)
%            Number of functors    :    7 (   7 usr;   1 con; 0-2 aty)
%            Number of variables   :  116 (   1 sgn;  74   !;   9   ?)

% Comments : 
%------------------------------------------------------------------------------
fof(f6,axiom,
    ! [X0,X1] : set_union2(X0,X1) = set_union2(X1,X0),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',commutativity_k2_xboole_0) ).

fof(f9,axiom,
    ! [X0,X1] : symmetric_difference(X0,X1) = set_union2(set_difference(X0,X1),set_difference(X1,X0)),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',d6_xboole_0) ).

fof(f20,axiom,
    ! [X0,X1] : set_union2(X0,X0) = X0,
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',idempotence_k2_xboole_0) ).

fof(f32,axiom,
    ! [X0,X1] : set_difference(X0,X1) = symmetric_difference(X0,set_intersection2(X0,X1)),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t100_xboole_1) ).

fof(f33,axiom,
    ! [X0] :
      ( preboolean(X0)
    <=> ! [X1,X2] :
          ( ( in(X2,X0)
            & in(X1,X0) )
         => ( in(set_difference(X1,X2),X0)
            & in(set_union2(X1,X2),X0) ) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t10_finsub_1) ).

fof(f34,conjecture,
    ! [X0] :
      ( ~ empty(X0)
     => ( ! [X1] :
            ( element(X1,X0)
           => ! [X2] :
                ( element(X2,X0)
               => ( in(set_union2(X1,X2),X0)
                  & in(symmetric_difference(X1,X2),X0) ) ) )
       => preboolean(X0) ) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t17_finsub_1) ).

fof(f35,negated_conjecture,
    ~ ! [X0] :
        ( ~ empty(X0)
       => ( ! [X1] :
              ( element(X1,X0)
             => ! [X2] :
                  ( element(X2,X0)
                 => ( in(set_union2(X1,X2),X0)
                    & in(symmetric_difference(X1,X2),X0) ) ) )
         => preboolean(X0) ) ),
    inference(negated_conjecture,[],[f34]) ).

fof(f37,axiom,
    ! [X0,X1] :
      ( in(X0,X1)
     => element(X0,X1) ),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t1_subset) ).

fof(f49,axiom,
    ! [X0,X1] : set_intersection2(X0,X1) = symmetric_difference(symmetric_difference(X0,X1),set_union2(X0,X1)),
    file('/export/starexec/sandbox2/benchmark/theBenchmark.p',t95_xboole_1) ).

fof(f50,plain,
    ! [X0] : set_union2(X0,X0) = X0,
    inference(rectify,[],[f20]) ).

fof(f80,plain,
    ! [X0] :
      ( preboolean(X0)
    <=> ! [X1,X2] :
          ( ( in(set_difference(X1,X2),X0)
            & in(set_union2(X1,X2),X0) )
          | ~ in(X2,X0)
          | ~ in(X1,X0) ) ),
    inference(ennf_transformation,[],[f33]) ).

fof(f81,plain,
    ! [X0] :
      ( preboolean(X0)
    <=> ! [X1,X2] :
          ( ( in(set_difference(X1,X2),X0)
            & in(set_union2(X1,X2),X0) )
          | ~ in(X2,X0)
          | ~ in(X1,X0) ) ),
    inference(flattening,[],[f80]) ).

fof(f82,plain,
    ? [X0] :
      ( ~ preboolean(X0)
      & ! [X1] :
          ( ! [X2] :
              ( ( in(set_union2(X1,X2),X0)
                & in(symmetric_difference(X1,X2),X0) )
              | ~ element(X2,X0) )
          | ~ element(X1,X0) )
      & ~ empty(X0) ),
    inference(ennf_transformation,[],[f35]) ).

fof(f83,plain,
    ? [X0] :
      ( ~ preboolean(X0)
      & ! [X1] :
          ( ! [X2] :
              ( ( in(set_union2(X1,X2),X0)
                & in(symmetric_difference(X1,X2),X0) )
              | ~ element(X2,X0) )
          | ~ element(X1,X0) )
      & ~ empty(X0) ),
    inference(flattening,[],[f82]) ).

fof(f84,plain,
    ! [X0,X1] :
      ( element(X0,X1)
      | ~ in(X0,X1) ),
    inference(ennf_transformation,[],[f37]) ).

fof(f114,plain,
    ! [X0] :
      ( ( preboolean(X0)
        | ? [X1,X2] :
            ( ( ~ in(set_difference(X1,X2),X0)
              | ~ in(set_union2(X1,X2),X0) )
            & in(X2,X0)
            & in(X1,X0) ) )
      & ( ! [X1,X2] :
            ( ( in(set_difference(X1,X2),X0)
              & in(set_union2(X1,X2),X0) )
            | ~ in(X2,X0)
            | ~ in(X1,X0) )
        | ~ preboolean(X0) ) ),
    inference(nnf_transformation,[],[f81]) ).

fof(f115,plain,
    ! [X0] :
      ( ( preboolean(X0)
        | ? [X1,X2] :
            ( ( ~ in(set_difference(X1,X2),X0)
              | ~ in(set_union2(X1,X2),X0) )
            & in(X2,X0)
            & in(X1,X0) ) )
      & ( ! [X3,X4] :
            ( ( in(set_difference(X3,X4),X0)
              & in(set_union2(X3,X4),X0) )
            | ~ in(X4,X0)
            | ~ in(X3,X0) )
        | ~ preboolean(X0) ) ),
    inference(rectify,[],[f114]) ).

fof(f116,plain,
    ! [X0] :
      ( ? [X1,X2] :
          ( ( ~ in(set_difference(X1,X2),X0)
            | ~ in(set_union2(X1,X2),X0) )
          & in(X2,X0)
          & in(X1,X0) )
     => ( ( ~ in(set_difference(sK10(X0),sK11(X0)),X0)
          | ~ in(set_union2(sK10(X0),sK11(X0)),X0) )
        & in(sK11(X0),X0)
        & in(sK10(X0),X0) ) ),
    introduced(choice_axiom,[]) ).

fof(f117,plain,
    ! [X0] :
      ( ( preboolean(X0)
        | ( ( ~ in(set_difference(sK10(X0),sK11(X0)),X0)
            | ~ in(set_union2(sK10(X0),sK11(X0)),X0) )
          & in(sK11(X0),X0)
          & in(sK10(X0),X0) ) )
      & ( ! [X3,X4] :
            ( ( in(set_difference(X3,X4),X0)
              & in(set_union2(X3,X4),X0) )
            | ~ in(X4,X0)
            | ~ in(X3,X0) )
        | ~ preboolean(X0) ) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f115,f116]) ).

fof(f118,plain,
    ( ? [X0] :
        ( ~ preboolean(X0)
        & ! [X1] :
            ( ! [X2] :
                ( ( in(set_union2(X1,X2),X0)
                  & in(symmetric_difference(X1,X2),X0) )
                | ~ element(X2,X0) )
            | ~ element(X1,X0) )
        & ~ empty(X0) )
   => ( ~ preboolean(sK12)
      & ! [X1] :
          ( ! [X2] :
              ( ( in(set_union2(X1,X2),sK12)
                & in(symmetric_difference(X1,X2),sK12) )
              | ~ element(X2,sK12) )
          | ~ element(X1,sK12) )
      & ~ empty(sK12) ) ),
    introduced(choice_axiom,[]) ).

fof(f119,plain,
    ( ~ preboolean(sK12)
    & ! [X1] :
        ( ! [X2] :
            ( ( in(set_union2(X1,X2),sK12)
              & in(symmetric_difference(X1,X2),sK12) )
            | ~ element(X2,sK12) )
        | ~ element(X1,sK12) )
    & ~ empty(sK12) ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK12])],[f83,f118]) ).

fof(f126,plain,
    ! [X0,X1] : set_union2(X0,X1) = set_union2(X1,X0),
    inference(cnf_transformation,[],[f6]) ).

fof(f129,plain,
    ! [X0,X1] : symmetric_difference(X0,X1) = set_union2(set_difference(X0,X1),set_difference(X1,X0)),
    inference(cnf_transformation,[],[f9]) ).

fof(f140,plain,
    ! [X0] : set_union2(X0,X0) = X0,
    inference(cnf_transformation,[],[f50]) ).

fof(f164,plain,
    ! [X0,X1] : set_difference(X0,X1) = symmetric_difference(X0,set_intersection2(X0,X1)),
    inference(cnf_transformation,[],[f32]) ).

fof(f167,plain,
    ! [X0] :
      ( preboolean(X0)
      | in(sK10(X0),X0) ),
    inference(cnf_transformation,[],[f117]) ).

fof(f168,plain,
    ! [X0] :
      ( preboolean(X0)
      | in(sK11(X0),X0) ),
    inference(cnf_transformation,[],[f117]) ).

fof(f169,plain,
    ! [X0] :
      ( preboolean(X0)
      | ~ in(set_difference(sK10(X0),sK11(X0)),X0)
      | ~ in(set_union2(sK10(X0),sK11(X0)),X0) ),
    inference(cnf_transformation,[],[f117]) ).

fof(f171,plain,
    ! [X2,X1] :
      ( in(symmetric_difference(X1,X2),sK12)
      | ~ element(X2,sK12)
      | ~ element(X1,sK12) ),
    inference(cnf_transformation,[],[f119]) ).

fof(f172,plain,
    ! [X2,X1] :
      ( in(set_union2(X1,X2),sK12)
      | ~ element(X2,sK12)
      | ~ element(X1,sK12) ),
    inference(cnf_transformation,[],[f119]) ).

fof(f173,plain,
    ~ preboolean(sK12),
    inference(cnf_transformation,[],[f119]) ).

fof(f175,plain,
    ! [X0,X1] :
      ( element(X0,X1)
      | ~ in(X0,X1) ),
    inference(cnf_transformation,[],[f84]) ).

fof(f187,plain,
    ! [X0,X1] : set_intersection2(X0,X1) = symmetric_difference(symmetric_difference(X0,X1),set_union2(X0,X1)),
    inference(cnf_transformation,[],[f49]) ).

fof(f188,plain,
    ! [X0,X1] : set_intersection2(X0,X1) = set_union2(set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1)),set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0)))),
    inference(definition_unfolding,[],[f187,f129,f129]) ).

fof(f189,plain,
    ! [X0,X1] : set_difference(X0,X1) = set_union2(set_difference(X0,set_union2(set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1)),set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0))))),set_difference(set_union2(set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1)),set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0)))),X0)),
    inference(definition_unfolding,[],[f164,f129,f188]) ).

fof(f196,plain,
    ! [X2,X1] :
      ( in(set_union2(set_difference(X1,X2),set_difference(X2,X1)),sK12)
      | ~ element(X2,sK12)
      | ~ element(X1,sK12) ),
    inference(definition_unfolding,[],[f171,f129]) ).

cnf(c_49,plain,
    set_union2(set_difference(X0,set_union2(set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1)),set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0))))),set_difference(set_union2(set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1)),set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0)))),X0)) = set_difference(X0,X1),
    inference(cnf_transformation,[],[f189]) ).

cnf(c_56,plain,
    set_union2(X0,X1) = set_union2(X1,X0),
    inference(cnf_transformation,[],[f126]) ).

cnf(c_69,plain,
    set_union2(X0,X0) = X0,
    inference(cnf_transformation,[],[f140]) ).

cnf(c_93,plain,
    ( ~ in(set_difference(sK10(X0),sK11(X0)),X0)
    | ~ in(set_union2(sK10(X0),sK11(X0)),X0)
    | preboolean(X0) ),
    inference(cnf_transformation,[],[f169]) ).

cnf(c_94,plain,
    ( in(sK11(X0),X0)
    | preboolean(X0) ),
    inference(cnf_transformation,[],[f168]) ).

cnf(c_95,plain,
    ( in(sK10(X0),X0)
    | preboolean(X0) ),
    inference(cnf_transformation,[],[f167]) ).

cnf(c_98,negated_conjecture,
    ~ preboolean(sK12),
    inference(cnf_transformation,[],[f173]) ).

cnf(c_99,negated_conjecture,
    ( ~ element(X0,sK12)
    | ~ element(X1,sK12)
    | in(set_union2(X0,X1),sK12) ),
    inference(cnf_transformation,[],[f172]) ).

cnf(c_100,negated_conjecture,
    ( ~ element(X0,sK12)
    | ~ element(X1,sK12)
    | in(set_union2(set_difference(X0,X1),set_difference(X1,X0)),sK12) ),
    inference(cnf_transformation,[],[f196]) ).

cnf(c_103,plain,
    ( ~ in(X0,X1)
    | element(X0,X1) ),
    inference(cnf_transformation,[],[f175]) ).

cnf(c_561,plain,
    set_union2(set_difference(X0,set_union2(set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0))),set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1)))),set_difference(set_union2(set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0))),set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1))),X0)) = set_difference(X0,X1),
    inference(demodulation,[status(thm)],[c_49,c_56]) ).

cnf(c_1651,plain,
    ( ~ element(X0,sK12)
    | in(X0,sK12) ),
    inference(superposition,[status(thm)],[c_69,c_99]) ).

cnf(c_1823,plain,
    ( ~ element(set_union2(set_difference(set_union2(X0,X1),set_union2(set_difference(X0,X1),set_difference(X1,X0))),set_difference(set_union2(set_difference(X0,X1),set_difference(X1,X0)),set_union2(X0,X1))),sK12)
    | ~ element(X0,sK12)
    | in(set_difference(X0,X1),sK12) ),
    inference(superposition,[status(thm)],[c_561,c_100]) ).

cnf(c_1892,plain,
    ( element(sK11(X0),X0)
    | preboolean(X0) ),
    inference(superposition,[status(thm)],[c_94,c_103]) ).

cnf(c_1893,plain,
    ( element(sK10(X0),X0)
    | preboolean(X0) ),
    inference(superposition,[status(thm)],[c_95,c_103]) ).

cnf(c_1894,plain,
    ( ~ element(X0,sK12)
    | ~ element(X1,sK12)
    | element(set_union2(X0,X1),sK12) ),
    inference(superposition,[status(thm)],[c_99,c_103]) ).

cnf(c_1895,plain,
    ( ~ element(X0,sK12)
    | ~ element(X1,sK12)
    | element(set_union2(set_difference(X0,X1),set_difference(X1,X0)),sK12) ),
    inference(superposition,[status(thm)],[c_100,c_103]) ).

cnf(c_2008,plain,
    ( ~ element(X0,sK12)
    | ~ element(X1,sK12)
    | element(set_union2(set_difference(X1,X0),set_difference(X0,X1)),sK12) ),
    inference(superposition,[status(thm)],[c_56,c_1895]) ).

cnf(c_2370,plain,
    ( ~ element(set_union2(set_difference(X0,X1),set_difference(X1,X0)),sK12)
    | ~ element(set_union2(X0,X1),sK12)
    | ~ element(X0,sK12)
    | in(set_difference(X0,X1),sK12) ),
    inference(superposition,[status(thm)],[c_2008,c_1823]) ).

cnf(c_2397,plain,
    ( ~ element(set_union2(X0,X1),sK12)
    | ~ element(X0,sK12)
    | ~ element(X1,sK12)
    | in(set_difference(X0,X1),sK12) ),
    inference(superposition,[status(thm)],[c_2008,c_2370]) ).

cnf(c_3874,plain,
    ( ~ element(X0,sK12)
    | ~ element(X1,sK12)
    | in(set_difference(X0,X1),sK12) ),
    inference(global_subsumption_just,[status(thm)],[c_2397,c_1894,c_2397]) ).

cnf(c_3889,plain,
    ( ~ in(set_union2(sK10(sK12),sK11(sK12)),sK12)
    | ~ element(sK10(sK12),sK12)
    | ~ element(sK11(sK12),sK12)
    | preboolean(sK12) ),
    inference(superposition,[status(thm)],[c_3874,c_93]) ).

cnf(c_3896,plain,
    ( ~ in(set_union2(sK10(sK12),sK11(sK12)),sK12)
    | ~ element(sK10(sK12),sK12)
    | ~ element(sK11(sK12),sK12) ),
    inference(forward_subsumption_resolution,[status(thm)],[c_3889,c_98]) ).

cnf(c_3915,plain,
    ( ~ element(sK10(sK12),sK12)
    | ~ element(sK11(sK12),sK12) ),
    inference(forward_subsumption_resolution,[status(thm)],[c_3896,c_99]) ).

cnf(c_4090,plain,
    ( in(sK11(sK12),sK12)
    | preboolean(sK12) ),
    inference(superposition,[status(thm)],[c_1892,c_1651]) ).

cnf(c_4092,plain,
    in(sK11(sK12),sK12),
    inference(forward_subsumption_resolution,[status(thm)],[c_4090,c_98]) ).

cnf(c_4125,plain,
    ( ~ element(sK11(sK12),sK12)
    | preboolean(sK12) ),
    inference(superposition,[status(thm)],[c_1893,c_3915]) ).

cnf(c_4127,plain,
    ~ element(sK11(sK12),sK12),
    inference(forward_subsumption_resolution,[status(thm)],[c_4125,c_98]) ).

cnf(c_4184,plain,
    element(sK11(sK12),sK12),
    inference(superposition,[status(thm)],[c_4092,c_103]) ).

cnf(c_4185,plain,
    $false,
    inference(prop_impl_just,[status(thm)],[c_4184,c_4127]) ).


%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12  % Problem  : SEU106+1 : TPTP v8.1.2. Released v3.2.0.
% 0.00/0.13  % Command  : run_iprover %s %d THM
% 0.12/0.34  % Computer : n021.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 : Thu Aug 24 01:05:25 EDT 2023
% 0.12/0.34  % CPUTime  : 
% 0.19/0.46  Running first-order theorem proving
% 0.19/0.46  Running: /export/starexec/sandbox2/solver/bin/run_problem --schedule fof_schedule --no_cores 8 /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 3.72/1.13  % SZS status Started for theBenchmark.p
% 3.72/1.13  % SZS status Theorem for theBenchmark.p
% 3.72/1.13  
% 3.72/1.13  %---------------- iProver v3.8 (pre SMT-COMP 2023/CASC 2023) ----------------%
% 3.72/1.13  
% 3.72/1.13  ------  iProver source info
% 3.72/1.13  
% 3.72/1.13  git: date: 2023-05-31 18:12:56 +0000
% 3.72/1.13  git: sha1: 8abddc1f627fd3ce0bcb8b4cbf113b3cc443d7b6
% 3.72/1.13  git: non_committed_changes: false
% 3.72/1.13  git: last_make_outside_of_git: false
% 3.72/1.13  
% 3.72/1.13  ------ Parsing...
% 3.72/1.13  ------ Clausification by vclausify_rel  & Parsing by iProver...
% 3.72/1.13  
% 3.72/1.13  ------ Preprocessing... sup_sim: 7  sf_s  rm: 13 0s  sf_e  pe_s  pe:1:0s pe:2:0s pe_e  sup_sim: 0  sf_s  rm: 4 0s  sf_e  pe_s  pe_e 
% 3.72/1.13  
% 3.72/1.13  ------ Preprocessing... gs_s  sp: 0 0s  gs_e  snvd_s sp: 0 0s snvd_e 
% 3.72/1.13  
% 3.72/1.13  ------ Preprocessing... sf_s  rm: 1 0s  sf_e  sf_s  rm: 0 0s  sf_e 
% 3.72/1.13  ------ Proving...
% 3.72/1.13  ------ Problem Properties 
% 3.72/1.13  
% 3.72/1.13  
% 3.72/1.13  clauses                                 46
% 3.72/1.13  conjectures                             4
% 3.72/1.13  EPR                                     14
% 3.72/1.13  Horn                                    40
% 3.72/1.13  unary                                   23
% 3.72/1.13  binary                                  14
% 3.72/1.13  lits                                    80
% 3.72/1.13  lits eq                                 10
% 3.72/1.13  fd_pure                                 0
% 3.72/1.13  fd_pseudo                               0
% 3.72/1.13  fd_cond                                 1
% 3.72/1.13  fd_pseudo_cond                          1
% 3.72/1.13  AC symbols                              0
% 3.72/1.13  
% 3.72/1.13  ------ Schedule dynamic 5 is on 
% 3.72/1.13  
% 3.72/1.13  ------ Input Options "--resolution_flag false --inst_lit_sel_side none" Time Limit: 10.
% 3.72/1.13  
% 3.72/1.13  
% 3.72/1.13  ------ 
% 3.72/1.13  Current options:
% 3.72/1.13  ------ 
% 3.72/1.13  
% 3.72/1.13  
% 3.72/1.13  
% 3.72/1.13  
% 3.72/1.13  ------ Proving...
% 3.72/1.13  
% 3.72/1.13  
% 3.72/1.13  % SZS status Theorem for theBenchmark.p
% 3.72/1.13  
% 3.72/1.13  % SZS output start CNFRefutation for theBenchmark.p
% See solution above
% 3.72/1.13  
% 3.72/1.13  
%------------------------------------------------------------------------------