I am trying to implement the incidence axioms in geometry for the Hilbert plane. And they came up with the following axioms:
interface (Eq point) => Plane line point where
-- Abstract notion for saying three points lie on the same line.
colinear : point -> point -> point -> Bool
coplanar : point -> point -> point -> Bool
contains : line -> point -> Bool
-- Intersection between two lines
intersects_at : line -> line -> point -> Bool
intersection_def : (contains l a = True) -> (contains m a = True) -> (intersects_at l m a = True)
-- For any two distinct points there is a line that contains them.
line_contains_two_points : (a,b : point) -> (a /= b) = True -> (l : line ** (contains l a = True, contains l b = True ))
-- If two points are contained by l and m then l = m
two_pts_define_line : contains l a = True -> contains l b = True -> contains m a = True -> contains m b = True -> l = m
-- There exists 3 non-colinear points.
three_non_colinear_pts : (a : point ** b : point ** c : point ** (colinear a b c = False, (a /= b) = True, (b /= c) = True, (a /= c) = True))
-- Any lines contains at least two points.
contain_two_pts : (l : line) -> (a : point ** b : point ** (contains l a = True, contains l b = True))
I want to show that a line crosses another line no more than once. So I came up with the following statement:
intersect_at_most_one_point : (l, m : line) -> (a : point) -> (intersects_at l m a = True) -> (intersects_at l m b = True) -> a = b
What is read:
Given two lines, if they intersect at two points aand bthen it should be that a = b.
However, I get an error message:
When checking type of Main.intersect_at_most_one_point:
When checking argument x to type constructor =:
Can't find implementation for Plane line point
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