TSTP Solution File: KRS157+1 by Z3---4.8.9.0

View Problem - Process Solution

%------------------------------------------------------------------------------
% File     : Z3---4.8.9.0
% Problem  : KRS157+1 : TPTP v8.1.0. Released v3.1.0.
% Transfm  : none
% Format   : tptp
% Command  : z3_tptp -proof -model -t:%d -file:%s

% Computer : n005.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 Sep 17 17:45:00 EDT 2022

% Result   : Theorem 0.20s 0.48s
% Output   : Proof 0.20s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :   10
%            Number of leaves      :   91
% Syntax   : Number of formulae    :  172 (  38 unt;  21 typ;   0 def)
%            Number of atoms       :  741 (   0 equ)
%            Maximal formula atoms :   24 (   4 avg)
%            Number of connectives : 1029 ( 451   ~; 191   |; 190   &)
%                                         ( 197 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   5 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of FOOLs       :   12 (  12 fml;   0 var)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   20 (  20   >;   0   *;   0   +;   0  <<)
%            Number of predicates  :   23 (  22 usr;   1 prp; 0-1 aty)
%            Number of functors    :    1 (   1 usr;   1 con; 0-0 aty)
%            Number of variables   :  111 (  99   !;   0   ?; 111   :)

% Comments : 
%------------------------------------------------------------------------------
tff(cC28_type,type,
    cC28: $i > $o ).

tff(iV3102_type,type,
    iV3102: $i ).

tff(cC14_type,type,
    cC14: $i > $o ).

tff(cC30_type,type,
    cC30: $i > $o ).

tff(cC26xcomp_type,type,
    cC26xcomp: $i > $o ).

tff(cC16xcomp_type,type,
    cC16xcomp: $i > $o ).

tff(cC2xcomp_type,type,
    cC2xcomp: $i > $o ).

tff(cC12xcomp_type,type,
    cC12xcomp: $i > $o ).

tff(cC216_type,type,
    cC216: $i > $o ).

tff(cC46_type,type,
    cC46: $i > $o ).

tff(cTEST_type,type,
    cTEST: $i > $o ).

tff(cC44_type,type,
    cC44: $i > $o ).

tff(cC42xcomp_type,type,
    cC42xcomp: $i > $o ).

tff(cC32xcomp_type,type,
    cC32xcomp: $i > $o ).

tff(cC64_type,type,
    cC64: $i > $o ).

tff(cTOP_type,type,
    cTOP: $i > $o ).

tff(cC62xcomp_type,type,
    cC62xcomp: $i > $o ).

tff(cowlThing_type,type,
    cowlThing: $i > $o ).

tff(xsd_integer_type,type,
    xsd_integer: $i > $o ).

tff(xsd_string_type,type,
    xsd_string: $i > $o ).

tff(cowlNothing_type,type,
    cowlNothing: $i > $o ).

tff(1,plain,
    ^ [X: $i] :
      refl(
        ( ( cC28(X)
        <=> ~ ( ~ cC16xcomp(X)
              | ~ cC26xcomp(X) ) )
      <=> ( cC28(X)
        <=> ~ ( ~ cC16xcomp(X)
              | ~ cC26xcomp(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(2,plain,
    ( ! [X: $i] :
        ( cC28(X)
      <=> ~ ( ~ cC16xcomp(X)
            | ~ cC26xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC28(X)
      <=> ~ ( ~ cC16xcomp(X)
            | ~ cC26xcomp(X) ) ) ),
    inference(quant_intro,[status(thm)],[1]) ).

tff(3,plain,
    ^ [X: $i] :
      rewrite(
        ( ( cC28(X)
        <=> ( cC16xcomp(X)
            & cC26xcomp(X) ) )
      <=> ( cC28(X)
        <=> ~ ( ~ cC16xcomp(X)
              | ~ cC26xcomp(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(4,plain,
    ( ! [X: $i] :
        ( cC28(X)
      <=> ( cC16xcomp(X)
          & cC26xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC28(X)
      <=> ~ ( ~ cC16xcomp(X)
            | ~ cC26xcomp(X) ) ) ),
    inference(quant_intro,[status(thm)],[3]) ).

tff(5,plain,
    ( ! [X: $i] :
        ( cC28(X)
      <=> ( cC16xcomp(X)
          & cC26xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC28(X)
      <=> ( cC16xcomp(X)
          & cC26xcomp(X) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(6,axiom,
    ! [X: $i] :
      ( cC28(X)
    <=> ( cC16xcomp(X)
        & cC26xcomp(X) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_131) ).

tff(7,plain,
    ! [X: $i] :
      ( cC28(X)
    <=> ( cC16xcomp(X)
        & cC26xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[6,5]) ).

tff(8,plain,
    ! [X: $i] :
      ( cC28(X)
    <=> ( cC16xcomp(X)
        & cC26xcomp(X) ) ),
    inference(skolemize,[status(sab)],[7]) ).

tff(9,plain,
    ! [X: $i] :
      ( cC28(X)
    <=> ~ ( ~ cC16xcomp(X)
          | ~ cC26xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[8,4]) ).

tff(10,plain,
    ! [X: $i] :
      ( cC28(X)
    <=> ~ ( ~ cC16xcomp(X)
          | ~ cC26xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[9,2]) ).

tff(11,plain,
    ( ~ ! [X: $i] :
          ( cC28(X)
        <=> ~ ( ~ cC16xcomp(X)
              | ~ cC26xcomp(X) ) )
    | ( cC28(iV3102)
    <=> ~ ( ~ cC16xcomp(iV3102)
          | ~ cC26xcomp(iV3102) ) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(12,plain,
    ( cC28(iV3102)
  <=> ~ ( ~ cC16xcomp(iV3102)
        | ~ cC26xcomp(iV3102) ) ),
    inference(unit_resolution,[status(thm)],[11,10]) ).

tff(13,plain,
    ( cC16xcomp(iV3102)
  <=> cC16xcomp(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(14,axiom,
    cC16xcomp(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_210) ).

tff(15,plain,
    cC16xcomp(iV3102),
    inference(modus_ponens,[status(thm)],[14,13]) ).

tff(16,plain,
    ( cC26xcomp(iV3102)
  <=> cC26xcomp(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(17,axiom,
    cC26xcomp(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_206) ).

tff(18,plain,
    cC26xcomp(iV3102),
    inference(modus_ponens,[status(thm)],[17,16]) ).

tff(19,plain,
    ( ~ ( ~ cC16xcomp(iV3102)
        | ~ cC26xcomp(iV3102) )
    | ~ cC16xcomp(iV3102)
    | ~ cC26xcomp(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(20,plain,
    ~ ( ~ cC16xcomp(iV3102)
      | ~ cC26xcomp(iV3102) ),
    inference(unit_resolution,[status(thm)],[19,18,15]) ).

tff(21,plain,
    ( ~ ( cC28(iV3102)
      <=> ~ ( ~ cC16xcomp(iV3102)
            | ~ cC26xcomp(iV3102) ) )
    | cC28(iV3102)
    | ~ cC16xcomp(iV3102)
    | ~ cC26xcomp(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(22,plain,
    ( ~ ( cC28(iV3102)
      <=> ~ ( ~ cC16xcomp(iV3102)
            | ~ cC26xcomp(iV3102) ) )
    | cC28(iV3102) ),
    inference(unit_resolution,[status(thm)],[21,20]) ).

tff(23,plain,
    cC28(iV3102),
    inference(unit_resolution,[status(thm)],[22,12]) ).

tff(24,plain,
    ^ [X: $i] :
      refl(
        ( ( cC14(X)
        <=> ~ ( ~ cC12xcomp(X)
              | ~ cC2xcomp(X) ) )
      <=> ( cC14(X)
        <=> ~ ( ~ cC12xcomp(X)
              | ~ cC2xcomp(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(25,plain,
    ( ! [X: $i] :
        ( cC14(X)
      <=> ~ ( ~ cC12xcomp(X)
            | ~ cC2xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC14(X)
      <=> ~ ( ~ cC12xcomp(X)
            | ~ cC2xcomp(X) ) ) ),
    inference(quant_intro,[status(thm)],[24]) ).

tff(26,plain,
    ^ [X: $i] :
      rewrite(
        ( ( cC14(X)
        <=> ( cC2xcomp(X)
            & cC12xcomp(X) ) )
      <=> ( cC14(X)
        <=> ~ ( ~ cC12xcomp(X)
              | ~ cC2xcomp(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(27,plain,
    ( ! [X: $i] :
        ( cC14(X)
      <=> ( cC2xcomp(X)
          & cC12xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC14(X)
      <=> ~ ( ~ cC12xcomp(X)
            | ~ cC2xcomp(X) ) ) ),
    inference(quant_intro,[status(thm)],[26]) ).

tff(28,plain,
    ( ! [X: $i] :
        ( cC14(X)
      <=> ( cC2xcomp(X)
          & cC12xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC14(X)
      <=> ( cC2xcomp(X)
          & cC12xcomp(X) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(29,axiom,
    ! [X: $i] :
      ( cC14(X)
    <=> ( cC2xcomp(X)
        & cC12xcomp(X) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_44) ).

tff(30,plain,
    ! [X: $i] :
      ( cC14(X)
    <=> ( cC2xcomp(X)
        & cC12xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[29,28]) ).

tff(31,plain,
    ! [X: $i] :
      ( cC14(X)
    <=> ( cC2xcomp(X)
        & cC12xcomp(X) ) ),
    inference(skolemize,[status(sab)],[30]) ).

tff(32,plain,
    ! [X: $i] :
      ( cC14(X)
    <=> ~ ( ~ cC12xcomp(X)
          | ~ cC2xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[31,27]) ).

tff(33,plain,
    ! [X: $i] :
      ( cC14(X)
    <=> ~ ( ~ cC12xcomp(X)
          | ~ cC2xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[32,25]) ).

tff(34,plain,
    ( ~ ! [X: $i] :
          ( cC14(X)
        <=> ~ ( ~ cC12xcomp(X)
              | ~ cC2xcomp(X) ) )
    | ( cC14(iV3102)
    <=> ~ ( ~ cC12xcomp(iV3102)
          | ~ cC2xcomp(iV3102) ) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(35,plain,
    ( cC14(iV3102)
  <=> ~ ( ~ cC12xcomp(iV3102)
        | ~ cC2xcomp(iV3102) ) ),
    inference(unit_resolution,[status(thm)],[34,33]) ).

tff(36,plain,
    ( cC12xcomp(iV3102)
  <=> cC12xcomp(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(37,axiom,
    cC12xcomp(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_215) ).

tff(38,plain,
    cC12xcomp(iV3102),
    inference(modus_ponens,[status(thm)],[37,36]) ).

tff(39,plain,
    ( cC2xcomp(iV3102)
  <=> cC2xcomp(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(40,axiom,
    cC2xcomp(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_203) ).

tff(41,plain,
    cC2xcomp(iV3102),
    inference(modus_ponens,[status(thm)],[40,39]) ).

tff(42,plain,
    ( ~ ( ~ cC12xcomp(iV3102)
        | ~ cC2xcomp(iV3102) )
    | ~ cC12xcomp(iV3102)
    | ~ cC2xcomp(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(43,plain,
    ~ ( ~ cC12xcomp(iV3102)
      | ~ cC2xcomp(iV3102) ),
    inference(unit_resolution,[status(thm)],[42,41,38]) ).

tff(44,plain,
    ( ~ ( cC14(iV3102)
      <=> ~ ( ~ cC12xcomp(iV3102)
            | ~ cC2xcomp(iV3102) ) )
    | cC14(iV3102)
    | ~ cC12xcomp(iV3102)
    | ~ cC2xcomp(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(45,plain,
    ( ~ ( cC14(iV3102)
      <=> ~ ( ~ cC12xcomp(iV3102)
            | ~ cC2xcomp(iV3102) ) )
    | cC14(iV3102) ),
    inference(unit_resolution,[status(thm)],[44,43]) ).

tff(46,plain,
    cC14(iV3102),
    inference(unit_resolution,[status(thm)],[45,35]) ).

tff(47,plain,
    ( ~ ( ~ cC14(iV3102)
        | ~ cC28(iV3102) )
    | ~ cC14(iV3102)
    | ~ cC28(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(48,plain,
    ~ ( ~ cC14(iV3102)
      | ~ cC28(iV3102) ),
    inference(unit_resolution,[status(thm)],[47,46,23]) ).

tff(49,plain,
    ^ [X: $i] :
      refl(
        ( ( cTEST(X)
        <=> ~ ( ~ cC216(X)
              | ~ cC46(X) ) )
      <=> ( cTEST(X)
        <=> ~ ( ~ cC216(X)
              | ~ cC46(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(50,plain,
    ( ! [X: $i] :
        ( cTEST(X)
      <=> ~ ( ~ cC216(X)
            | ~ cC46(X) ) )
  <=> ! [X: $i] :
        ( cTEST(X)
      <=> ~ ( ~ cC216(X)
            | ~ cC46(X) ) ) ),
    inference(quant_intro,[status(thm)],[49]) ).

tff(51,plain,
    ^ [X: $i] :
      rewrite(
        ( ( cTEST(X)
        <=> ( cC216(X)
            & cC46(X) ) )
      <=> ( cTEST(X)
        <=> ~ ( ~ cC216(X)
              | ~ cC46(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(52,plain,
    ( ! [X: $i] :
        ( cTEST(X)
      <=> ( cC216(X)
          & cC46(X) ) )
  <=> ! [X: $i] :
        ( cTEST(X)
      <=> ~ ( ~ cC216(X)
            | ~ cC46(X) ) ) ),
    inference(quant_intro,[status(thm)],[51]) ).

tff(53,plain,
    ( ! [X: $i] :
        ( cTEST(X)
      <=> ( cC216(X)
          & cC46(X) ) )
  <=> ! [X: $i] :
        ( cTEST(X)
      <=> ( cC216(X)
          & cC46(X) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(54,axiom,
    ! [X: $i] :
      ( cTEST(X)
    <=> ( cC216(X)
        & cC46(X) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_201) ).

tff(55,plain,
    ! [X: $i] :
      ( cTEST(X)
    <=> ( cC216(X)
        & cC46(X) ) ),
    inference(modus_ponens,[status(thm)],[54,53]) ).

tff(56,plain,
    ! [X: $i] :
      ( cTEST(X)
    <=> ( cC216(X)
        & cC46(X) ) ),
    inference(skolemize,[status(sab)],[55]) ).

tff(57,plain,
    ! [X: $i] :
      ( cTEST(X)
    <=> ~ ( ~ cC216(X)
          | ~ cC46(X) ) ),
    inference(modus_ponens,[status(thm)],[56,52]) ).

tff(58,plain,
    ! [X: $i] :
      ( cTEST(X)
    <=> ~ ( ~ cC216(X)
          | ~ cC46(X) ) ),
    inference(modus_ponens,[status(thm)],[57,50]) ).

tff(59,plain,
    ( ~ ! [X: $i] :
          ( cTEST(X)
        <=> ~ ( ~ cC216(X)
              | ~ cC46(X) ) )
    | ( cTEST(iV3102)
    <=> ~ ( ~ cC216(iV3102)
          | ~ cC46(iV3102) ) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(60,plain,
    ( cTEST(iV3102)
  <=> ~ ( ~ cC216(iV3102)
        | ~ cC46(iV3102) ) ),
    inference(unit_resolution,[status(thm)],[59,58]) ).

tff(61,plain,
    ( cTEST(iV3102)
  <=> cTEST(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(62,axiom,
    cTEST(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_202) ).

tff(63,plain,
    cTEST(iV3102),
    inference(modus_ponens,[status(thm)],[62,61]) ).

tff(64,plain,
    ( ~ ( cTEST(iV3102)
      <=> ~ ( ~ cC216(iV3102)
            | ~ cC46(iV3102) ) )
    | ~ cTEST(iV3102)
    | ~ ( ~ cC216(iV3102)
        | ~ cC46(iV3102) ) ),
    inference(tautology,[status(thm)],]) ).

tff(65,plain,
    ( ~ ( cTEST(iV3102)
      <=> ~ ( ~ cC216(iV3102)
            | ~ cC46(iV3102) ) )
    | ~ ( ~ cC216(iV3102)
        | ~ cC46(iV3102) ) ),
    inference(unit_resolution,[status(thm)],[64,63]) ).

tff(66,plain,
    ~ ( ~ cC216(iV3102)
      | ~ cC46(iV3102) ),
    inference(unit_resolution,[status(thm)],[65,60]) ).

tff(67,plain,
    ( ~ cC216(iV3102)
    | ~ cC46(iV3102)
    | cC216(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(68,plain,
    cC216(iV3102),
    inference(unit_resolution,[status(thm)],[67,66]) ).

tff(69,plain,
    ^ [X: $i] :
      refl(
        ( ( cC44(X)
        <=> ~ ( ~ cC32xcomp(X)
              | ~ cC42xcomp(X) ) )
      <=> ( cC44(X)
        <=> ~ ( ~ cC32xcomp(X)
              | ~ cC42xcomp(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(70,plain,
    ( ! [X: $i] :
        ( cC44(X)
      <=> ~ ( ~ cC32xcomp(X)
            | ~ cC42xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC44(X)
      <=> ~ ( ~ cC32xcomp(X)
            | ~ cC42xcomp(X) ) ) ),
    inference(quant_intro,[status(thm)],[69]) ).

tff(71,plain,
    ^ [X: $i] :
      rewrite(
        ( ( cC44(X)
        <=> ( cC42xcomp(X)
            & cC32xcomp(X) ) )
      <=> ( cC44(X)
        <=> ~ ( ~ cC32xcomp(X)
              | ~ cC42xcomp(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(72,plain,
    ( ! [X: $i] :
        ( cC44(X)
      <=> ( cC42xcomp(X)
          & cC32xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC44(X)
      <=> ~ ( ~ cC32xcomp(X)
            | ~ cC42xcomp(X) ) ) ),
    inference(quant_intro,[status(thm)],[71]) ).

tff(73,plain,
    ( ! [X: $i] :
        ( cC44(X)
      <=> ( cC42xcomp(X)
          & cC32xcomp(X) ) )
  <=> ! [X: $i] :
        ( cC44(X)
      <=> ( cC42xcomp(X)
          & cC32xcomp(X) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(74,axiom,
    ! [X: $i] :
      ( cC44(X)
    <=> ( cC42xcomp(X)
        & cC32xcomp(X) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_147) ).

tff(75,plain,
    ! [X: $i] :
      ( cC44(X)
    <=> ( cC42xcomp(X)
        & cC32xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[74,73]) ).

tff(76,plain,
    ! [X: $i] :
      ( cC44(X)
    <=> ( cC42xcomp(X)
        & cC32xcomp(X) ) ),
    inference(skolemize,[status(sab)],[75]) ).

tff(77,plain,
    ! [X: $i] :
      ( cC44(X)
    <=> ~ ( ~ cC32xcomp(X)
          | ~ cC42xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[76,72]) ).

tff(78,plain,
    ! [X: $i] :
      ( cC44(X)
    <=> ~ ( ~ cC32xcomp(X)
          | ~ cC42xcomp(X) ) ),
    inference(modus_ponens,[status(thm)],[77,70]) ).

tff(79,plain,
    ( ~ ! [X: $i] :
          ( cC44(X)
        <=> ~ ( ~ cC32xcomp(X)
              | ~ cC42xcomp(X) ) )
    | ( cC44(iV3102)
    <=> ~ ( ~ cC32xcomp(iV3102)
          | ~ cC42xcomp(iV3102) ) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(80,plain,
    ( cC44(iV3102)
  <=> ~ ( ~ cC32xcomp(iV3102)
        | ~ cC42xcomp(iV3102) ) ),
    inference(unit_resolution,[status(thm)],[79,78]) ).

tff(81,plain,
    ( cC42xcomp(iV3102)
  <=> cC42xcomp(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(82,axiom,
    cC42xcomp(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_211) ).

tff(83,plain,
    cC42xcomp(iV3102),
    inference(modus_ponens,[status(thm)],[82,81]) ).

tff(84,plain,
    ( cC32xcomp(iV3102)
  <=> cC32xcomp(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(85,axiom,
    cC32xcomp(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_207) ).

tff(86,plain,
    cC32xcomp(iV3102),
    inference(modus_ponens,[status(thm)],[85,84]) ).

tff(87,plain,
    ( ~ ( ~ cC32xcomp(iV3102)
        | ~ cC42xcomp(iV3102) )
    | ~ cC32xcomp(iV3102)
    | ~ cC42xcomp(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(88,plain,
    ~ ( ~ cC32xcomp(iV3102)
      | ~ cC42xcomp(iV3102) ),
    inference(unit_resolution,[status(thm)],[87,86,83]) ).

tff(89,plain,
    ( ~ ( cC44(iV3102)
      <=> ~ ( ~ cC32xcomp(iV3102)
            | ~ cC42xcomp(iV3102) ) )
    | cC44(iV3102)
    | ~ cC32xcomp(iV3102)
    | ~ cC42xcomp(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(90,plain,
    ( ~ ( cC44(iV3102)
      <=> ~ ( ~ cC32xcomp(iV3102)
            | ~ cC42xcomp(iV3102) ) )
    | cC44(iV3102) ),
    inference(unit_resolution,[status(thm)],[89,88]) ).

tff(91,plain,
    cC44(iV3102),
    inference(unit_resolution,[status(thm)],[90,80]) ).

tff(92,plain,
    ^ [X: $i] :
      refl(
        ( ( cC64(X)
        <=> ~ ( ~ cC62xcomp(X)
              | ~ cTOP(X) ) )
      <=> ( cC64(X)
        <=> ~ ( ~ cC62xcomp(X)
              | ~ cTOP(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(93,plain,
    ( ! [X: $i] :
        ( cC64(X)
      <=> ~ ( ~ cC62xcomp(X)
            | ~ cTOP(X) ) )
  <=> ! [X: $i] :
        ( cC64(X)
      <=> ~ ( ~ cC62xcomp(X)
            | ~ cTOP(X) ) ) ),
    inference(quant_intro,[status(thm)],[92]) ).

tff(94,plain,
    ^ [X: $i] :
      rewrite(
        ( ( cC64(X)
        <=> ( cC62xcomp(X)
            & cTOP(X) ) )
      <=> ( cC64(X)
        <=> ~ ( ~ cC62xcomp(X)
              | ~ cTOP(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(95,plain,
    ( ! [X: $i] :
        ( cC64(X)
      <=> ( cC62xcomp(X)
          & cTOP(X) ) )
  <=> ! [X: $i] :
        ( cC64(X)
      <=> ~ ( ~ cC62xcomp(X)
            | ~ cTOP(X) ) ) ),
    inference(quant_intro,[status(thm)],[94]) ).

tff(96,plain,
    ( ! [X: $i] :
        ( cC64(X)
      <=> ( cC62xcomp(X)
          & cTOP(X) ) )
  <=> ! [X: $i] :
        ( cC64(X)
      <=> ( cC62xcomp(X)
          & cTOP(X) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(97,axiom,
    ! [X: $i] :
      ( cC64(X)
    <=> ( cC62xcomp(X)
        & cTOP(X) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_166) ).

tff(98,plain,
    ! [X: $i] :
      ( cC64(X)
    <=> ( cC62xcomp(X)
        & cTOP(X) ) ),
    inference(modus_ponens,[status(thm)],[97,96]) ).

tff(99,plain,
    ! [X: $i] :
      ( cC64(X)
    <=> ( cC62xcomp(X)
        & cTOP(X) ) ),
    inference(skolemize,[status(sab)],[98]) ).

tff(100,plain,
    ! [X: $i] :
      ( cC64(X)
    <=> ~ ( ~ cC62xcomp(X)
          | ~ cTOP(X) ) ),
    inference(modus_ponens,[status(thm)],[99,95]) ).

tff(101,plain,
    ! [X: $i] :
      ( cC64(X)
    <=> ~ ( ~ cC62xcomp(X)
          | ~ cTOP(X) ) ),
    inference(modus_ponens,[status(thm)],[100,93]) ).

tff(102,plain,
    ( ~ ! [X: $i] :
          ( cC64(X)
        <=> ~ ( ~ cC62xcomp(X)
              | ~ cTOP(X) ) )
    | ( cC64(iV3102)
    <=> ~ ( ~ cC62xcomp(iV3102)
          | ~ cTOP(iV3102) ) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(103,plain,
    ( cC64(iV3102)
  <=> ~ ( ~ cC62xcomp(iV3102)
        | ~ cTOP(iV3102) ) ),
    inference(unit_resolution,[status(thm)],[102,101]) ).

tff(104,plain,
    ( cC62xcomp(iV3102)
  <=> cC62xcomp(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(105,axiom,
    cC62xcomp(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_212) ).

tff(106,plain,
    cC62xcomp(iV3102),
    inference(modus_ponens,[status(thm)],[105,104]) ).

tff(107,plain,
    ( cTOP(iV3102)
  <=> cTOP(iV3102) ),
    inference(rewrite,[status(thm)],]) ).

tff(108,axiom,
    cTOP(iV3102),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_205) ).

tff(109,plain,
    cTOP(iV3102),
    inference(modus_ponens,[status(thm)],[108,107]) ).

tff(110,plain,
    ( ~ ( ~ cC62xcomp(iV3102)
        | ~ cTOP(iV3102) )
    | ~ cC62xcomp(iV3102)
    | ~ cTOP(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(111,plain,
    ~ ( ~ cC62xcomp(iV3102)
      | ~ cTOP(iV3102) ),
    inference(unit_resolution,[status(thm)],[110,109,106]) ).

tff(112,plain,
    ( ~ ( cC64(iV3102)
      <=> ~ ( ~ cC62xcomp(iV3102)
            | ~ cTOP(iV3102) ) )
    | cC64(iV3102)
    | ~ cC62xcomp(iV3102)
    | ~ cTOP(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(113,plain,
    ( ~ ( cC64(iV3102)
      <=> ~ ( ~ cC62xcomp(iV3102)
            | ~ cTOP(iV3102) ) )
    | cC64(iV3102) ),
    inference(unit_resolution,[status(thm)],[112,111]) ).

tff(114,plain,
    cC64(iV3102),
    inference(unit_resolution,[status(thm)],[113,103]) ).

tff(115,plain,
    ( ~ cC216(iV3102)
    | ~ cC46(iV3102)
    | cC46(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(116,plain,
    cC46(iV3102),
    inference(unit_resolution,[status(thm)],[115,66]) ).

tff(117,plain,
    ( ~ ~ ( ~ cC14(iV3102)
          | ~ cC44(iV3102)
          | ~ cC216(iV3102)
          | ~ cC64(iV3102)
          | ~ cC46(iV3102)
          | ~ cC28(iV3102)
          | ~ cC30(iV3102) )
  <=> ( ~ cC14(iV3102)
      | ~ cC44(iV3102)
      | ~ cC216(iV3102)
      | ~ cC64(iV3102)
      | ~ cC46(iV3102)
      | ~ cC28(iV3102)
      | ~ cC30(iV3102) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(118,plain,
    ( ( cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cC30(iV3102) )
  <=> ~ ( ~ cC14(iV3102)
        | ~ cC44(iV3102)
        | ~ cC216(iV3102)
        | ~ cC64(iV3102)
        | ~ cC46(iV3102)
        | ~ cC28(iV3102)
        | ~ cC30(iV3102) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(119,plain,
    ( ~ ( cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cC30(iV3102) )
  <=> ~ ~ ( ~ cC14(iV3102)
          | ~ cC44(iV3102)
          | ~ cC216(iV3102)
          | ~ cC64(iV3102)
          | ~ cC46(iV3102)
          | ~ cC28(iV3102)
          | ~ cC30(iV3102) ) ),
    inference(monotonicity,[status(thm)],[118]) ).

tff(120,plain,
    ( ~ ( cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cC30(iV3102) )
  <=> ( ~ cC14(iV3102)
      | ~ cC44(iV3102)
      | ~ cC216(iV3102)
      | ~ cC64(iV3102)
      | ~ cC46(iV3102)
      | ~ cC28(iV3102)
      | ~ cC30(iV3102) ) ),
    inference(transitivity,[status(thm)],[119,117]) ).

tff(121,plain,
    ( ~ ( cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cC30(iV3102) )
  <=> ~ ( cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cC30(iV3102) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(122,plain,
    ( ~ ( ! [X: $i] :
            ( cowlThing(X)
            & ~ cowlNothing(X) )
        & ! [X: $i] :
            ( xsd_string(X)
          <=> ~ xsd_integer(X) )
        & cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cowlThing(iV3102)
        & cC30(iV3102) )
  <=> ~ ( cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cC30(iV3102) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(123,plain,
    ( ( ! [X: $i] :
          ( cowlThing(X)
          & ~ cowlNothing(X) )
      & ! [X: $i] :
          ( xsd_string(X)
        <=> ~ xsd_integer(X) )
      & cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cowlThing(iV3102)
      & cC30(iV3102) )
  <=> ( ! [X: $i] :
          ( cowlThing(X)
          & ~ cowlNothing(X) )
      & ! [X: $i] :
          ( xsd_string(X)
        <=> ~ xsd_integer(X) )
      & cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cowlThing(iV3102)
      & cC30(iV3102) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(124,plain,
    ( ~ ( ! [X: $i] :
            ( cowlThing(X)
            & ~ cowlNothing(X) )
        & ! [X: $i] :
            ( xsd_string(X)
          <=> ~ xsd_integer(X) )
        & cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cowlThing(iV3102)
        & cC30(iV3102) )
  <=> ~ ( ! [X: $i] :
            ( cowlThing(X)
            & ~ cowlNothing(X) )
        & ! [X: $i] :
            ( xsd_string(X)
          <=> ~ xsd_integer(X) )
        & cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cowlThing(iV3102)
        & cC30(iV3102) ) ),
    inference(monotonicity,[status(thm)],[123]) ).

tff(125,plain,
    ( ~ ( ! [X: $i] :
            ( cowlThing(X)
            & ~ cowlNothing(X) )
        & ! [X: $i] :
            ( xsd_string(X)
          <=> ~ xsd_integer(X) )
        & cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cowlThing(iV3102)
        & cC30(iV3102) )
  <=> ~ ( ! [X: $i] :
            ( cowlThing(X)
            & ~ cowlNothing(X) )
        & ! [X: $i] :
            ( xsd_string(X)
          <=> ~ xsd_integer(X) )
        & cC14(iV3102)
        & cC44(iV3102)
        & cC216(iV3102)
        & cC64(iV3102)
        & cC46(iV3102)
        & cC28(iV3102)
        & cowlThing(iV3102)
        & cC30(iV3102) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(126,axiom,
    ~ ( ! [X: $i] :
          ( cowlThing(X)
          & ~ cowlNothing(X) )
      & ! [X: $i] :
          ( xsd_string(X)
        <=> ~ xsd_integer(X) )
      & cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cowlThing(iV3102)
      & cC30(iV3102) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',the_axiom) ).

tff(127,plain,
    ~ ( ! [X: $i] :
          ( cowlThing(X)
          & ~ cowlNothing(X) )
      & ! [X: $i] :
          ( xsd_string(X)
        <=> ~ xsd_integer(X) )
      & cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cowlThing(iV3102)
      & cC30(iV3102) ),
    inference(modus_ponens,[status(thm)],[126,125]) ).

tff(128,plain,
    ~ ( ! [X: $i] :
          ( cowlThing(X)
          & ~ cowlNothing(X) )
      & ! [X: $i] :
          ( xsd_string(X)
        <=> ~ xsd_integer(X) )
      & cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cowlThing(iV3102)
      & cC30(iV3102) ),
    inference(modus_ponens,[status(thm)],[127,124]) ).

tff(129,plain,
    ~ ( cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cC30(iV3102) ),
    inference(modus_ponens,[status(thm)],[128,122]) ).

tff(130,plain,
    ~ ( cC14(iV3102)
      & cC44(iV3102)
      & cC216(iV3102)
      & cC64(iV3102)
      & cC46(iV3102)
      & cC28(iV3102)
      & cC30(iV3102) ),
    inference(modus_ponens,[status(thm)],[129,121]) ).

tff(131,plain,
    ( ~ cC14(iV3102)
    | ~ cC44(iV3102)
    | ~ cC216(iV3102)
    | ~ cC64(iV3102)
    | ~ cC46(iV3102)
    | ~ cC28(iV3102)
    | ~ cC30(iV3102) ),
    inference(modus_ponens,[status(thm)],[130,120]) ).

tff(132,plain,
    ~ cC30(iV3102),
    inference(unit_resolution,[status(thm)],[131,116,46,114,91,23,68]) ).

tff(133,plain,
    ( ~ ( cC30(iV3102)
      <=> ~ ( ~ cC14(iV3102)
            | ~ cC28(iV3102) ) )
    | cC30(iV3102)
    | ~ cC14(iV3102)
    | ~ cC28(iV3102) ),
    inference(tautology,[status(thm)],]) ).

tff(134,plain,
    ~ ( cC30(iV3102)
    <=> ~ ( ~ cC14(iV3102)
          | ~ cC28(iV3102) ) ),
    inference(unit_resolution,[status(thm)],[133,132,48]) ).

tff(135,plain,
    ^ [X: $i] :
      refl(
        ( ( cC30(X)
        <=> ~ ( ~ cC28(X)
              | ~ cC14(X) ) )
      <=> ( cC30(X)
        <=> ~ ( ~ cC28(X)
              | ~ cC14(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(136,plain,
    ( ! [X: $i] :
        ( cC30(X)
      <=> ~ ( ~ cC28(X)
            | ~ cC14(X) ) )
  <=> ! [X: $i] :
        ( cC30(X)
      <=> ~ ( ~ cC28(X)
            | ~ cC14(X) ) ) ),
    inference(quant_intro,[status(thm)],[135]) ).

tff(137,plain,
    ^ [X: $i] :
      rewrite(
        ( ( cC30(X)
        <=> ( cC28(X)
            & cC14(X) ) )
      <=> ( cC30(X)
        <=> ~ ( ~ cC28(X)
              | ~ cC14(X) ) ) )),
    inference(bind,[status(th)],]) ).

tff(138,plain,
    ( ! [X: $i] :
        ( cC30(X)
      <=> ( cC28(X)
          & cC14(X) ) )
  <=> ! [X: $i] :
        ( cC30(X)
      <=> ~ ( ~ cC28(X)
            | ~ cC14(X) ) ) ),
    inference(quant_intro,[status(thm)],[137]) ).

tff(139,plain,
    ( ! [X: $i] :
        ( cC30(X)
      <=> ( cC28(X)
          & cC14(X) ) )
  <=> ! [X: $i] :
        ( cC30(X)
      <=> ( cC28(X)
          & cC14(X) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(140,axiom,
    ! [X: $i] :
      ( cC30(X)
    <=> ( cC28(X)
        & cC14(X) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',axiom_132) ).

tff(141,plain,
    ! [X: $i] :
      ( cC30(X)
    <=> ( cC28(X)
        & cC14(X) ) ),
    inference(modus_ponens,[status(thm)],[140,139]) ).

tff(142,plain,
    ! [X: $i] :
      ( cC30(X)
    <=> ( cC28(X)
        & cC14(X) ) ),
    inference(skolemize,[status(sab)],[141]) ).

tff(143,plain,
    ! [X: $i] :
      ( cC30(X)
    <=> ~ ( ~ cC28(X)
          | ~ cC14(X) ) ),
    inference(modus_ponens,[status(thm)],[142,138]) ).

tff(144,plain,
    ! [X: $i] :
      ( cC30(X)
    <=> ~ ( ~ cC28(X)
          | ~ cC14(X) ) ),
    inference(modus_ponens,[status(thm)],[143,136]) ).

tff(145,plain,
    ( ( ~ ! [X: $i] :
            ( cC30(X)
          <=> ~ ( ~ cC28(X)
                | ~ cC14(X) ) )
      | ( cC30(iV3102)
      <=> ~ ( ~ cC14(iV3102)
            | ~ cC28(iV3102) ) ) )
  <=> ( ~ ! [X: $i] :
            ( cC30(X)
          <=> ~ ( ~ cC28(X)
                | ~ cC14(X) ) )
      | ( cC30(iV3102)
      <=> ~ ( ~ cC14(iV3102)
            | ~ cC28(iV3102) ) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(146,plain,
    ( ( cC30(iV3102)
    <=> ~ ( ~ cC28(iV3102)
          | ~ cC14(iV3102) ) )
  <=> ( cC30(iV3102)
    <=> ~ ( ~ cC14(iV3102)
          | ~ cC28(iV3102) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(147,plain,
    ( ( ~ ! [X: $i] :
            ( cC30(X)
          <=> ~ ( ~ cC28(X)
                | ~ cC14(X) ) )
      | ( cC30(iV3102)
      <=> ~ ( ~ cC28(iV3102)
            | ~ cC14(iV3102) ) ) )
  <=> ( ~ ! [X: $i] :
            ( cC30(X)
          <=> ~ ( ~ cC28(X)
                | ~ cC14(X) ) )
      | ( cC30(iV3102)
      <=> ~ ( ~ cC14(iV3102)
            | ~ cC28(iV3102) ) ) ) ),
    inference(monotonicity,[status(thm)],[146]) ).

tff(148,plain,
    ( ( ~ ! [X: $i] :
            ( cC30(X)
          <=> ~ ( ~ cC28(X)
                | ~ cC14(X) ) )
      | ( cC30(iV3102)
      <=> ~ ( ~ cC28(iV3102)
            | ~ cC14(iV3102) ) ) )
  <=> ( ~ ! [X: $i] :
            ( cC30(X)
          <=> ~ ( ~ cC28(X)
                | ~ cC14(X) ) )
      | ( cC30(iV3102)
      <=> ~ ( ~ cC14(iV3102)
            | ~ cC28(iV3102) ) ) ) ),
    inference(transitivity,[status(thm)],[147,145]) ).

tff(149,plain,
    ( ~ ! [X: $i] :
          ( cC30(X)
        <=> ~ ( ~ cC28(X)
              | ~ cC14(X) ) )
    | ( cC30(iV3102)
    <=> ~ ( ~ cC28(iV3102)
          | ~ cC14(iV3102) ) ) ),
    inference(quant_inst,[status(thm)],]) ).

tff(150,plain,
    ( ~ ! [X: $i] :
          ( cC30(X)
        <=> ~ ( ~ cC28(X)
              | ~ cC14(X) ) )
    | ( cC30(iV3102)
    <=> ~ ( ~ cC14(iV3102)
          | ~ cC28(iV3102) ) ) ),
    inference(modus_ponens,[status(thm)],[149,148]) ).

tff(151,plain,
    $false,
    inference(unit_resolution,[status(thm)],[150,144,134]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12  % Problem  : KRS157+1 : TPTP v8.1.0. Released v3.1.0.
% 0.07/0.13  % Command  : z3_tptp -proof -model -t:%d -file:%s
% 0.13/0.34  % Computer : n005.cluster.edu
% 0.13/0.34  % Model    : x86_64 x86_64
% 0.13/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34  % Memory   : 8042.1875MB
% 0.13/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34  % CPULimit : 300
% 0.13/0.34  % WCLimit  : 300
% 0.13/0.34  % DateTime : Thu Sep  1 09:39:48 EDT 2022
% 0.13/0.34  % CPUTime  : 
% 0.13/0.34  Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.13/0.34  Usage: tptp [options] [-file:]file
% 0.13/0.34    -h, -?       prints this message.
% 0.13/0.34    -smt2        print SMT-LIB2 benchmark.
% 0.13/0.34    -m, -model   generate model.
% 0.13/0.34    -p, -proof   generate proof.
% 0.13/0.34    -c, -core    generate unsat core of named formulas.
% 0.13/0.34    -st, -statistics display statistics.
% 0.13/0.34    -t:timeout   set timeout (in second).
% 0.13/0.34    -smt2status  display status in smt2 format instead of SZS.
% 0.13/0.34    -check_status check the status produced by Z3 against annotation in benchmark.
% 0.13/0.34    -<param>:<value> configuration parameter and value.
% 0.13/0.34    -o:<output-file> file to place output in.
% 0.20/0.48  % SZS status Theorem
% 0.20/0.48  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------