TPTP Axioms File: QUA001^1.ax


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% File     : QUA001^1 : TPTP v7.5.0. Released v4.1.0.
% Domain   : Quantales
% Axioms   : Tests for Quantales (Boolean sub-algebra below 1)
% Version  : [Hoe09] axioms.
% English  :

% Refs     : [Con71] Conway (1971), Regular Algebra and Finite Machines
%          : [Koz97] Kozen (1997), Kleene Algebra with Tests
%          : [Hoe09] Hoefner (2009), Email to Geoff Sutcliffe
% Source   : [Hoe09]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :    2 (   0 unit;   1 type;   0 defn)
%            Number of atoms       :   21 (   3 equality;   7 variable)
%            Maximal formula depth :    9 (   6 average)
%            Number of connectives :   10 (   0   ~;   0   |;   2   &;   7   @)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&;   0  !!;   0  ??)
%            Number of type conns  :    1 (   1   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   1   :;   0  :=)
%            Number of variables   :    2 (   0 sgn;   1   !;   1   ?;   0   ^)
%                                         (   2   :;   0  :=;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : 

% Comments : Requires QUA001^0.ax
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thf(tests,type,(
    test: $i > $o )).

thf(test_definition,axiom,(
    ! [X: $i] :
      ( ( test @ X )
     => ? [Y: $i] :
          ( ( ( addition @ X @ Y )
            = one )
          & ( ( multiplication @ X @ Y )
            = zero )
          & ( ( multiplication @ Y @ X )
            = zero ) ) ) )).

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