## TPTP Axioms File: HEN001-0.ax

```%--------------------------------------------------------------------------
% File     : HEN001-0 : 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   : [MOW76]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses    :    9 (   0 non-Horn;   3 unit;   6 RR)
%            Number of atoms      :   21 (   2 equality)
%            Maximal clause size  :    6 (   2 average)
%            Number of predicates :    3 (   0 propositional; 2-3 arity)
%            Number of functors   :    3 (   2 constant; 0-2 arity)
%            Number of variables  :   25 (   3 singleton)
%            Maximal term depth   :    2 (   1 average)
% SPC      :

%--------------------------------------------------------------------------
%----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 )).

%--------------------------------------------------------------------------
```