英文:
Point-free function to add 2 elements to the list / double application of (:) list data constructor and (.)
问题
我正在努力正确定义函数的无点版本，该函数将两个元素添加到列表中。
很容易想出一些直接的简单实现：
addTwoElems :: a -> a -> [a] -> [a]
addTwoElems x y xs = x : y : xs
addTwoElems x y = (++) [x, y]
addTwoElems = (.) `on` (:) -- 无点版本，但使用了额外的函数
但是，如何使用两个列表数据构造函数 (:) 进行无点组合 (.) 呢？
请不仅展示正确的函数实现，还请解释每个步骤和逻辑，说明如何得出正确的版本。
英文:
I am struggling to define correctly the point-free version of the function, which adds 2 elements to the list.
It is easy to come up with a number of straightforward trivial implmentations:
addTwoElems :: a -> a -> [a] -> [a]
addTwoElems x y xs = x : y : xs
addTwoElems x y = (++) [x, y]
addTwoElems = (.) `on` (:) “ point free but with additional function
But how would a point-free composition (.) of the two list data constructors (:) look like?
Please not just show the correct function implementation, but please with explanations of the steps and logic behind how to come to the right version.
答案1
得分: 6
根据评论，以下是仅使用 . 和 : 的逐步推导过程：
addTwoElems x y xs = x : y : xs
-- 将第一个 : 重写为前缀函数
addTwoElems x y xs = (:) x (y : xs)
-- 将第二个 : 重写为前缀函数
addTwoElems x y xs = (:) x ((:) y xs)
-- 使用函数组合将 xs 从括号中提取出来
addTwoElems x y xs = ((:) x . (:) y) xs
-- 使用 eta 缩减来消除 xs
addTwoElems x y = (:) x . (:) y
-- 将 . 重写为前缀函数
addTwoElems x y = (.) ((:) x) ((:) y)
-- 使用函数组合将 y 从括号中提取出来
addTwoElems x y = ((.) ((:) x) . (:)) y
-- 使用 eta 缩减来消除 y
addTwoElems x = (.) ((:) x) . (:)
-- 将第二个 . 重写为操作符部分，使得表达式中包含 x 的部分位于最后
addTwoElems x = (. (:)) ((.) ((:) x))
-- 使用函数组合将 x 从内部括号中提取出来
addTwoElems x = (. (:)) (((.) . (:)) x)
-- 使用函数组合将 x 从外部括号中提取出来
addTwoElems x = ((. (:)) . (.) . (:)) x
-- 使用 eta 缩减来消除 x
addTwoElems = (. (:)) . (.) . (:)
以上是给定代码的逐步推导过程。
英文:
Per the comments, a step-by-step derivation that only uses . and ::
addTwoElems x y xs = x : y : xs
-- rewrite the first : as a prefix function
addTwoElems x y xs = (:) x (y : xs)
-- rewrite the second : as a prefix function
addTwoElems x y xs = (:) x ((:) y xs)
-- use function composition to get xs out of the parentheses
addTwoElems x y xs = ((:) x . (:) y) xs
-- eta-reduce to get rid of xs
addTwoElems x y = (:) x . (:) y
-- rewrite the . as a prefix function
addTwoElems x y = (.) ((:) x) ((:) y)
-- use function composition to get y out of the parentheses
addTwoElems x y = ((.) ((:) x) . (:)) y
-- eta-reduce to get rid of y
addTwoElems x = (.) ((:) x) . (:)
-- rewrite the second . as an operator section, so that the part of the expression with x is last
addTwoElems x = (. (:)) ((.) ((:) x))
-- use function composition to get x out of the inner parentheses
addTwoElems x = (. (:)) (((.) . (:)) x)
-- use function composition to get x out of the outer parentheses
addTwoElems x = ((. (:)) . (.) . (:)) x
-- eta-reduce to get rid of x
addTwoElems = (. (:)) . (.) . (:)
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