Question

我一直在尝试在Hilbert平面的几何中实现入射公理。并想出了以下公理：

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))

我想表明一条线最多与另一条线相交一次。所以我想出了以下声明：

intersect_at_most_one_point : (l, m : line) -> (a : point) -> (intersects_at l m a = True) -> (intersects_at l m b = True) -> a = b

其中包括：

给定两行，如果它们在两个点a和b相交，那么它必须是a = b。

然而我收到错误：

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

所以我怀疑这意味着它需要某种data值，我可以证明它满足入射几何的想法。我在数学上解释这个，因为我需要一个系统模型。问题是有很多几何形状和＃34;满足这些完全不同的公理。

是否有可能推导出有关接口的定理而无需使用任何明确的data？

Answer 1

您需要将Plane约束添加到intersect_at_most_one_point的类型签名：

intersect_at_most_one_point : Plane line point => 
  (l, m : line) -> (a : point) -> 
  (intersects_at l m a = True) -> (intersects_at l m b = True) -> 
  a = b

有没有办法在没有模型的情况下证明伊德里斯的东西？

1 个答案: