%------------------------------------------------------------------------------
% File     : DAT002=1 : TPTP v7.2.0. Released v5.0.0.
% Domain   : Data Structures
% Axioms   : Integer collections with counting
% Version  : [Wal10] axioms.
% English  : 

% Refs     : [PW06]  Prevosto & Waldmann (2006), SPASS+T
%          : [Wal10] Waldmann (2010), Email to Geoff Sutcliffe
% Source   : [Wal10]
% Names    : 

% Status   : Satisfiable
% Syntax   : Number of formulae    :    8 (   1 unit;   1 type)
%            Number of atoms       :   13 (   7 equality)
%            Maximal formula depth :    5 (   4 average)
%            Number of connectives :    8 (   2   ~;   0   |;   0   &)
%                                         (   5 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    1 (   1   >;   0   *;   0   +;   0  <<)
%            Number of predicates  :    6 (   3 propositional; 0-2 arity)
%            Number of functors    :    8 (   3 constant; 0-2 arity)
%            Number of variables   :   12 (   0 sgn;  12   !;   0   ?)
%                                         (  12   :;   0  !>;   0  ?*)
%            Maximal term depth    :    3 (   2 average)
%            Arithmetic symbols    :   10 (   1 prd;   2 fun;   2 num;   5 var)
% SPC      : TFF_SAT_RFO_SEQ_SAR

% Comments : Requires DAT002=0
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tff(count_type,type,(
    count: collection > $int )).

tff(ax1,axiom,(
    ! [X6: collection] : $greatereq(count(X6),0) )).

tff(ax2,axiom,(
    ! [X7: collection] :
      ( X7 = empty
    <=> count(X7) = 0 ) )).

tff(ax3,axiom,(
    ! [X8: $int,X9: collection] :
      ( ~ in(X8,X9)
    <=> count(add(X8,X9)) = $sum(count(X9),1) ) )).

tff(ax4,axiom,(
    ! [X10: $int,X11: collection] :
      ( in(X10,X11)
    <=> count(add(X10,X11)) = count(X11) ) )).

tff(ax5,axiom,(
    ! [X12: $int,X13: collection] :
      ( in(X12,X13)
    <=> count(remove(X12,X13)) = $difference(count(X13),1) ) )).

tff(ax6,axiom,(
    ! [X14: $int,X15: collection] :
      ( ~ in(X14,X15)
    <=> count(remove(X14,X15)) = count(X15) ) )).

tff(ax7,axiom,(
    ! [X16: $int,X17: collection] :
      ( in(X16,X17)
     => X17 = add(X16,remove(X16,X17)) ) )).

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