TPTP Axioms File: GEO005-0.ax
%--------------------------------------------------------------------------
% File : GEO005-0 : TPTP v9.0.0. Released v2.7.0.
% Domain : Geometry (Hilbert)
% Axioms : Hilbert geometry axioms, adapted to respect multi-sortedness
% Version : [Cla03] axioms.
% English :
% Refs : [Ben92] Benana992), Recognising Unnecessary Clauses in Res
% : [Cla03] Claessen (2003), Email to G. Sutcliffe
% Source : [Cla03]
% Names :
% Status : Satisfiable
% Syntax : Number of clauses : 31 ( 1 unt; 18 nHn; 31 RR)
% Number of literals : 174 ( 43 equ; 103 neg)
% Maximal clause size : 16 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-3 aty)
% Number of functors : 10 ( 10 usr; 1 con; 0-3 aty)
% Number of variables : 70 ( 0 sgn)
% SPC :
% Comments :
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%----Axiom 1 : For any two distinct points, there is a unique line through
%----them.
cnf(axiom_G1A,axiom,
( point_on_line(Z1,line_from_to(Z1,Z2))
| Z1 = Z2
| ~ point(Z1)
| ~ point(Z2) ) ).
cnf(axiom_G1B,axiom,
( point_on_line(Z2,line_from_to(Z1,Z2))
| Z1 = Z2
| ~ point(Z1)
| ~ point(Z2) ) ).
cnf(axiom_G1C,axiom,
( line(line_from_to(Z1,Z2))
| Z1 = Z2
| ~ point(Z1)
| ~ point(Z2) ) ).
cnf(axiom_G1D,axiom,
( ~ point_on_line(Z1,Y3)
| Z1 = Z2
| ~ point_on_line(Z2,Y3)
| Y3 = Y4
| ~ point_on_line(Z1,Y4)
| ~ point_on_line(Z2,Y4)
| ~ point(Z1)
| ~ point(Z2)
| ~ line(Y3)
| ~ line(Y4) ) ).
%----For any line, there are at least two points on the line.
cnf(axiom_G2A,axiom,
( point_on_line(point_1_on_line(Y1),Y1)
| ~ line(Y1) ) ).
cnf(axiom_G2B,axiom,
( point_on_line(point_2_on_line(Y1),Y1)
| ~ line(Y1) ) ).
cnf(axiom_G2C,axiom,
( point(point_1_on_line(Y1))
| ~ line(Y1) ) ).
cnf(axiom_G2D,axiom,
( point(point_2_on_line(Y1))
| ~ line(Y1) ) ).
cnf(axiom_G2E,axiom,
( point_1_on_line(Y1) != point_2_on_line(Y1)
| ~ line(Y1) ) ).
%----For any line, there is a point not on the line.
cnf(axiom_G3A,axiom,
( ~ point_on_line(a_point_not_on_line(Y1),Y1)
| ~ line(Y1) ) ).
cnf(axiom_G3B,axiom,
( point(a_point_not_on_line(Y1))
| ~ line(Y1) ) ).
%----There exists at least one line
cnf(axiom_G4A,axiom,
line(at_least_one_line) ).
%----For any plane there is a point on the plane.
cnf(axiom_G5A,axiom,
( ~ plane(Z1)
| point_on_plane(a_point_on_plane(Z1),Z1) ) ).
cnf(axiom_G5B,axiom,
( ~ plane(Z1)
| point(a_point_on_plane(Z1)) ) ).
%----For any plane there is a point not on the plane.
cnf(axiom_G6A,axiom,
( ~ plane(Z1)
| ~ point_on_plane(a_point_not_on_plane(Z1),Z1) ) ).
cnf(axiom_G6B,axiom,
( ~ plane(Z1)
| point(a_point_not_on_plane(Z1)) ) ).
%----For any three non-collinear points there is a unique plane through
%----them.
cnf(axiom_G7A,axiom,
( point_on_plane(X1,plane_for_points(X1,X2,X3))
| ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| collinear(X1,X2,X3)
| X1 = X2
| X1 = X3
| X2 = X3 ) ).
cnf(axiom_G7B,axiom,
( point_on_plane(X2,plane_for_points(X1,X2,X3))
| ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| collinear(X1,X2,X3)
| X1 = X2
| X1 = X3
| X2 = X3 ) ).
cnf(axiom_G7C,axiom,
( point_on_plane(X3,plane_for_points(X1,X2,X3))
| ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| collinear(X1,X2,X3)
| X1 = X2
| X1 = X3
| X2 = X3 ) ).
cnf(axiom_G7D,axiom,
( plane(plane_for_points(X1,X2,X3))
| ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| collinear(X1,X2,X3)
| X1 = X2
| X1 = X3
| X2 = X3 ) ).
cnf(axiom_G7E,axiom,
( ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| collinear(X1,X2,X3)
| X1 = X2
| X1 = X3
| X2 = X3
| ~ point_on_plane(X1,Z1)
| ~ point_on_plane(X2,Z1)
| ~ point_on_plane(X3,Z1)
| ~ plane(Z1)
| ~ point_on_plane(X1,Z2)
| ~ point_on_plane(X2,Z2)
| ~ point_on_plane(X3,Z2)
| ~ plane(Z2)
| Z1 = Z2 ) ).
%----If two points of a line are in the same plane then every point
%----of that line is in the plane.
cnf(axiom_G8A,axiom,
( ~ point_on_line(X1,Y1)
| ~ point_on_line(X2,Y1)
| ~ point_on_plane(X1,Z1)
| ~ point_on_plane(X2,Z1)
| ~ plane(Z1)
| ~ point(X1)
| ~ point(X2)
| ~ line(Y1)
| X1 = X2
| line_on_plane(Y1,Z1) ) ).
%----If two planes have a point in common they have at least one more
%----point in common.
cnf(axiom_G9A,axiom,
( ~ plane(Z1)
| ~ plane(Z2)
| Z1 = Z2
| ~ point_on_plane(X1,Z1)
| ~ point_on_plane(X1,Z2)
| ~ point(X1)
| point_on_plane(common_point_on_planes(Z1,Z2,X1),Z1) ) ).
cnf(axiom_G9B,axiom,
( ~ plane(Z1)
| ~ plane(Z2)
| Z1 = Z2
| ~ point_on_plane(X1,Z1)
| ~ point_on_plane(X1,Z2)
| ~ point(X1)
| point_on_plane(common_point_on_planes(Z1,Z2,X1),Z2) ) ).
cnf(axiom_G9C,axiom,
( ~ plane(Z1)
| ~ plane(Z2)
| Z1 = Z2
| ~ point_on_plane(X1,Z1)
| ~ point_on_plane(X1,Z2)
| ~ point(X1)
| point(common_point_on_planes(Z1,Z2,X1)) ) ).
cnf(axiom_G9D,axiom,
( ~ plane(Z1)
| ~ plane(Z2)
| Z1 = Z2
| ~ point_on_plane(X1,Z1)
| ~ point_on_plane(X1,Z2)
| ~ point(X1)
| X1 != common_point_on_planes(Z1,Z2,X1) ) ).
%----Three distinct points are collinear if and only if there is a line
%----through them.
cnf(axiom_G10A,axiom,
( ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| X1 = X2
| X1 = X3
| X2 = X3
| point_on_line(X1,line_through_3_points(X1,X2,X3))
| ~ collinear(X1,X2,X3) ) ).
cnf(axiom_G10B,axiom,
( ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| X1 = X2
| X1 = X3
| X2 = X3
| point_on_line(X2,line_through_3_points(X1,X2,X3))
| ~ collinear(X1,X2,X3) ) ).
cnf(axiom_G10C,axiom,
( ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| X1 = X2
| X1 = X3
| X2 = X3
| point_on_line(X3,line_through_3_points(X1,X2,X3))
| ~ collinear(X1,X2,X3) ) ).
cnf(axiom_G10D,axiom,
( ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| X1 = X2
| X1 = X3
| X2 = X3
| line(line_through_3_points(X1,X2,X3))
| ~ collinear(X1,X2,X3) ) ).
cnf(axiom_G10E,axiom,
( collinear(X1,X2,X3)
| ~ point_on_line(X1,Y)
| ~ point_on_line(X2,Y)
| ~ point_on_line(X3,Y)
| ~ point(X1)
| ~ point(X2)
| ~ point(X3)
| X1 = X2
| X1 = X3
| X2 = X3
| ~ line(Y) ) ).
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