TPTP Problem File: MSC031^1.p

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%------------------------------------------------------------------------------
% File     : MSC031^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Miscellaneous (Pigeon-hole principle)
% Problem  : International Mathematical Olympiad, 1972, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Prove that from a set of ten distinct two-digit numbers (in the 
%            decimal system), it is possible to select two disjoint subsets 
%            whose members have the same sum.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1972-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45402 (2213 equality;22745 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39694 ( 104   ~; 233   |;1187   &;36043   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8074 (  66 sgn;7096   !; 433   ?; 409   ^)
%                                         (8074   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1989 (   6 prd;   9 fun;  24 num;1950 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: PA; Score: 5; Author: Takuya Matsuzaki;
%            Generated: 2015-01-12
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
    ! [V_n0: $int,V_n1: $int,V_n2: $int,V_n3: $int,V_n4: $int,V_n5: $int,V_n6: $int,V_n7: $int,V_n8: $int,V_n9: $int,V_ns: ( 'ListOf' @ $int )] :
      ( ( ( V_ns
          = ( 'cons/2' @ $int @ V_n0 @ ( 'cons/2' @ $int @ V_n1 @ ( 'cons/2' @ $int @ V_n2 @ ( 'cons/2' @ $int @ V_n3 @ ( 'cons/2' @ $int @ V_n4 @ ( 'cons/2' @ $int @ V_n5 @ ( 'cons/2' @ $int @ V_n6 @ ( 'cons/2' @ $int @ V_n7 @ ( 'cons/2' @ $int @ V_n8 @ ( 'cons/2' @ $int @ V_n9 @ ( 'nil/0' @ $int ) ) ) ) ) ) ) ) ) ) ) )
        & ( 'all/2' @ $int
          @ ^ [V_m: $int] :
              ( ( $lesseq @ 10 @ V_m )
              & ( $lesseq @ V_m @ 99 ) )
          @ V_ns )
        & ( $less @ V_n0 @ V_n1 )
        & ( $less @ V_n1 @ V_n2 )
        & ( $less @ V_n2 @ V_n3 )
        & ( $less @ V_n3 @ V_n4 )
        & ( $less @ V_n4 @ V_n5 )
        & ( $less @ V_n5 @ V_n6 )
        & ( $less @ V_n6 @ V_n7 )
        & ( $less @ V_n7 @ V_n8 )
        & ( $less @ V_n8 @ V_n9 ) )
     => ? [V_ns1: ( 'ListOf' @ $int ),V_ns2: ( 'ListOf' @ $int ),V_ss1: ( 'SetOf' @ $int ),V_ss2: ( 'SetOf' @ $int )] :
          ( ( V_ss1
            = ( 'set-by-def/1' @ $int
              @ ^ [V_m_dot_2: $int] :
                  ( 'member/2' @ $int @ V_m_dot_2 @ V_ns1 ) ) )
          & ( V_ss2
            = ( 'set-by-def/1' @ $int
              @ ^ [V_m_dot_1: $int] :
                  ( 'member/2' @ $int @ V_m_dot_1 @ V_ns2 ) ) )
          & ( 'is-empty/1' @ $int @ ( 'set-intersection/2' @ $int @ V_ss1 @ V_ss2 ) )
          & ( ( 'set-union/2' @ $int @ V_ss1 @ V_ss2 )
            = ( 'set-by-def/1' @ $int
              @ ^ [V_m_dot_0: $int] :
                  ( 'member/2' @ $int @ V_m_dot_0 @ V_ns ) ) )
          & ( ( 'int.sum/1' @ V_ns1 )
            = ( 'int.sum/1' @ V_ns2 ) ) ) ) )).

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