%--------------------------------------------------------------------------
% File     : HEN001-0 : TPTP v9.0.0. Released v1.0.0.
% Domain   : Henkin Models
% Axioms   : Henkin model axioms
% Version  : [MOW76] axioms.
% English  :

% Refs     : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source   : [MOW76]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses     :    9 (   3 unt;   0 nHn;   6 RR)
%            Number of literals    :   21 (   2 equ;  12 neg)
%            Maximal clause size   :    6 (   2 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 2-3 aty)
%            Number of functors    :    3 (   3 usr;   2 con; 0-2 aty)
%            Number of variables   :   25 (   3 sgn)
% SPC      : 

% Comments :
%--------------------------------------------------------------------------
%----A0: definition of less than or equal to
cnf(quotient_less_equal,axiom,
    ( ~ less_equal(X,Y)
    | quotient(X,Y,zero) ) ).

cnf(less_equal_quotient,axiom,
    ( ~ quotient(X,Y,zero)
    | less_equal(X,Y) ) ).

%----A1: x/y <= x
cnf(divisor_existence,axiom,
    ( ~ quotient(X,Y,Z)
    | less_equal(Z,X) ) ).

%----A2: (x/z) / (y/z) <= (x/y) / z
cnf(quotient_property,axiom,
    ( ~ quotient(X,Y,V1)
    | ~ quotient(Y,Z,V2)
    | ~ quotient(X,Z,V3)
    | ~ quotient(V3,V2,V4)
    | ~ quotient(V1,Z,V5)
    | less_equal(V4,V5) ) ).

%----A3: 0 <= x
cnf(zero_is_smallest,axiom,
    less_equal(zero,X) ).

%----A4: x <= y and y <= x implies that x = y
cnf(less_equal_and_equal,axiom,
    ( ~ less_equal(X,Y)
    | ~ less_equal(Y,X)
    | X = Y ) ).

%----A5: x <= identity (Thus an implicative model with unit 1)
cnf(identity_is_largest,axiom,
    less_equal(X,identity) ).

%----closure of '/'
cnf(closure,axiom,
    quotient(X,Y,divide(X,Y)) ).

%----'/' is well defined
cnf(well_defined,axiom,
    ( ~ quotient(X,Y,Z)
    | ~ quotient(X,Y,W)
    | Z = W ) ).

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