## TPTP Axioms File: HEN002-0.ax

```%--------------------------------------------------------------------------
% File     : HEN002-0 : TPTP v7.5.0. 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   : [ANL]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :    7 (   0 non-Horn;   4 unit;   3 RR)
%            Number of atoms      :   11 (   3 equality)
%            Maximal clause size  :    3 (   2 average)
%            Number of predicates :    2 (   0 propositional; 2-2 arity)
%            Number of functors   :    3 (   2 constant; 0-2 arity)
%            Number of variables  :   13 (   3 singleton)
%            Maximal term depth   :    3 (   1 average)
% SPC      :

%--------------------------------------------------------------------------
%----A0: Definition of less_equal
cnf(quotient_less_equal1,axiom,
( ~ less_equal(X,Y)
| divide(X,Y) = zero )).

cnf(quotient_less_equal2,axiom,
( divide(X,Y) != zero
| less_equal(X,Y) )).

%----A1: x/y <= x
cnf(quotient_smaller_than_numerator,axiom,
( less_equal(divide(X,Y),X) )).

%----A2: (x/z) / (y/z) <= (x/y) / z
cnf(quotient_property,axiom,
( less_equal(divide(divide(X,Z),divide(Y,Z)),divide(divide(X,Y),Z)) )).

%----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 )
cnf(identity_is_largest,axiom,
( less_equal(X,identity) )).

%----Implicit in equality formulation: '/' is well defined
%--------------------------------------------------------------------------
```