## TPTP Axioms File: GEO008+0.ax

```%------------------------------------------------------------------------------
% File     : GEO008+0 : TPTP v7.5.0. Released v3.3.0.
% Domain   : Geometry (Constructive)
% Axioms   : Apartness geometry
% Version  : [Li97] axioms.
% English  :

% Refs     : [Li98]  Li (1998), A Shorter and Intuitive Axiom to Replace th
%          : [Li97]  Li (1997), Replacing the Axioms for Connecting Lines a
%          : [vPl95] von Plato (1995), The Axioms of Constructive Geometry
% Source   : [ILTP]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of formulae    :   12 (   3 unit)
%            Number of atoms       :   34 (   0 equality)
%            Maximal formula depth :    9 (   6 average)
%            Number of connectives :   25 (   3 ~  ;   9  |;   2  &)
%                                         (   0 <=>;  11 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    4 (   0 propositional; 2-2 arity)
%            Number of functors    :    2 (   0 constant; 2-2 arity)
%            Number of variables   :   30 (   0 singleton;  30 !;   0 ?)
%            Maximal term depth    :    2 (   1 average)
% SPC      :

%------------------------------------------------------------------------------
%----Apartness for distinct points, distinct lines, convergent lines
fof(apart1,axiom,(
! [X] : ~ distinct_points(X,X) )).

fof(apart2,axiom,(
! [X] : ~ distinct_lines(X,X) )).

fof(apart3,axiom,(
! [X] : ~ convergent_lines(X,X) )).

fof(apart4,axiom,(
! [X,Y,Z] :
( distinct_points(X,Y)
=> ( distinct_points(X,Z)
| distinct_points(Y,Z) ) ) )).

fof(apart5,axiom,(
! [X,Y,Z] :
( distinct_lines(X,Y)
=> ( distinct_lines(X,Z)
| distinct_lines(Y,Z) ) ) )).

fof(apart6,axiom,(
! [X,Y,Z] :
( convergent_lines(X,Y)
=> ( convergent_lines(X,Z)
| convergent_lines(Y,Z) ) ) )).

%----Connecting lines and intersection points
fof(con1,axiom,(
! [X,Y,Z] :
( distinct_points(X,Y)
=> ( apart_point_and_line(Z,line_connecting(X,Y))
=> ( distinct_points(Z,X)
& distinct_points(Z,Y) ) ) ) )).

fof(con2,axiom,(
! [X,Y,Z] :
( convergent_lines(X,Y)
=> ( ( apart_point_and_line(Z,X)
| apart_point_and_line(Z,Y) )
=> distinct_points(Z,intersection_point(X,Y)) ) ) )).

%----Constructive uniqueness axiom for lines and points
fof(cu1,axiom,(
! [X,Y,U,V] :
( ( distinct_points(X,Y)
& distinct_lines(U,V) )
=> ( apart_point_and_line(X,U)
| apart_point_and_line(X,V)
| apart_point_and_line(Y,U)
| apart_point_and_line(Y,V) ) ) )).

%----Compatibility of equality with apartness and convergence
fof(ceq1,axiom,(
! [X,Y,Z] :
( apart_point_and_line(X,Y)
=> ( distinct_points(X,Z)
| apart_point_and_line(Z,Y) ) ) )).

fof(ceq2,axiom,(
! [X,Y,Z] :
( apart_point_and_line(X,Y)
=> ( distinct_lines(Y,Z)
| apart_point_and_line(X,Z) ) ) )).

fof(ceq3,axiom,(
! [X,Y] :
( convergent_lines(X,Y)
=> distinct_lines(X,Y) ) )).

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