%------------------------------------------------------------------------------
% File     : Z3---4.8.9.0
% Problem  : ARI599_1 : TPTP v8.1.0. Released v5.1.0.
% Transfm  : none
% Format   : tptp
% Command  : z3_tptp -proof -model -t:%d -file:%s

% Computer : n029.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 17:02:13 EDT 2022

% Result   : Theorem 0.18s 0.37s
% Output   : Proof 0.18s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    9
%            Number of leaves      :    7
% Syntax   : Number of formulae    :   19 (  10 unt;   3 typ;   0 def)
%            Number of atoms       :   28 (   9 equ)
%            Maximal formula atoms :    6 (   1 avg)
%            Number of connectives :   19 (   7   ~;   2   |;   5   &)
%                                         (   3 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   2 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number arithmetic     :   90 (  18 atm;  36 fun;  36 num;   0 var)
%            Number of types       :    1 (   0 usr;   1 ari)
%            Number of type conns  :    2 (   1   >;   1   *;   0   +;   0  <<)
%            Number of predicates  :    4 (   0 usr;   1 prp; 0-2 aty)
%            Number of functors    :    9 (   3 usr;   6 con; 0-2 aty)
%            Number of variables   :    0 (   0   !;   0   ?;   0   :)

% Comments : 
%------------------------------------------------------------------------------
tff(f_type,type,
    f: ( $int * $int ) > $int ).

tff(a_type,type,
    a: $int ).

tff(b_type,type,
    b: $int ).

tff(1,plain,
    ( $lesseq($product(2,a),$product(2,b))
  <=> $lesseq($sum(a,$product(-1,b)),0) ),
    inference(rewrite,[status(thm)],]) ).

tff(2,plain,
    ( ~ ( ( $lesseq($product(2,a),$product(2,b))
          & $lesseq($product(3,b),$product(3,a)) )
       => ( f(a,b) = f(b,a) ) )
  <=> ~ ( ~ ( $lesseq($product(2,a),$product(2,b))
            & $lesseq($product(3,b),$product(3,a)) )
        | ( f(a,b) = f(b,a) ) ) ),
    inference(rewrite,[status(thm)],]) ).

tff(3,axiom,
    ~ ( ( $lesseq($product(2,a),$product(2,b))
        & $lesseq($product(3,b),$product(3,a)) )
     => ( f(a,b) = f(b,a) ) ),
    file('/export/starexec/sandbox/benchmark/theBenchmark.p',ineq_imply_f_eq_f) ).

tff(4,plain,
    ~ ( ~ ( $lesseq($product(2,a),$product(2,b))
          & $lesseq($product(3,b),$product(3,a)) )
      | ( f(a,b) = f(b,a) ) ),
    inference(modus_ponens,[status(thm)],[3,2]) ).

tff(5,plain,
    ( $lesseq($product(2,a),$product(2,b))
    & $lesseq($product(3,b),$product(3,a)) ),
    inference(or_elim,[status(thm)],[4]) ).

tff(6,plain,
    $lesseq($product(2,a),$product(2,b)),
    inference(and_elim,[status(thm)],[5]) ).

tff(7,plain,
    $lesseq($sum(a,$product(-1,b)),0),
    inference(modus_ponens,[status(thm)],[6,1]) ).

tff(8,plain,
    ( $lesseq($product(3,b),$product(3,a))
  <=> $greatereq($sum(a,$product(-1,b)),0) ),
    inference(rewrite,[status(thm)],]) ).

tff(9,plain,
    $lesseq($product(3,b),$product(3,a)),
    inference(and_elim,[status(thm)],[5]) ).

tff(10,plain,
    $greatereq($sum(a,$product(-1,b)),0),
    inference(modus_ponens,[status(thm)],[9,8]) ).

tff(11,plain,
    a = b,
    inference(theory_lemma,[status(thm)],[10,7]) ).

tff(12,plain,
    b = a,
    inference(symmetry,[status(thm)],[11]) ).

tff(13,plain,
    f(b,a) = f(a,b),
    inference(monotonicity,[status(thm)],[12,11]) ).

tff(14,plain,
    f(a,b) = f(b,a),
    inference(symmetry,[status(thm)],[13]) ).

tff(15,plain,
    f(a,b) != f(b,a),
    inference(or_elim,[status(thm)],[4]) ).

tff(16,plain,
    $false,
    inference(unit_resolution,[status(thm)],[15,14]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12  % Problem  : ARI599_1 : TPTP v8.1.0. Released v5.1.0.
% 0.03/0.13  % Command  : z3_tptp -proof -model -t:%d -file:%s
% 0.12/0.33  % Computer : n029.cluster.edu
% 0.12/0.33  % Model    : x86_64 x86_64
% 0.12/0.33  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33  % Memory   : 8042.1875MB
% 0.12/0.33  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33  % CPULimit : 300
% 0.12/0.33  % WCLimit  : 300
% 0.12/0.33  % DateTime : Tue Aug 30 00:52:41 EDT 2022
% 0.12/0.33  % 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.
% 0.18/0.37  % SZS status Theorem
% 0.18/0.37  % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------