Question

我在看How does inorder+preorder construct unique binary tree?，并认为在伊德里斯写一个正式的证据会很有趣。不幸的是，我很早就陷入了困境，试图证明在树中找到元素的方法与在顺序遍历中找到它的方式相对应（当然，我还需要为预先排序做到这一点）遍历）。任何想法都会受到欢迎。我对完整的解决方案并不特别感兴趣 - 更多的是帮助我们开始正确的方向。

鉴于

data Tree a = Tip
            | Node (Tree a) a (Tree a)

我可以通过至少两种方式将其转换为列表：

inorder : Tree a -> List a
inorder Tip = []
inorder (Node l v r) = inorder l ++ [v] ++ inorder r

或

foldrTree : (a -> b -> b) -> b -> Tree a -> b
foldrTree c n Tip = n
foldrTree c n (Node l v r) = foldr c (v `c` foldrTree c n r) l
inorder = foldrTree (::) []

第二种方法似乎使所有很难，所以我的大部分努力都集中在第一种方法上。我在树中描述这样的位置：

data InTree : a -> Tree a -> Type where
  AtRoot : x `InTree` Node l x r
  OnLeft : x `InTree` l -> x `InTree` Node l v r
  OnRight : x `InTree` r -> x `InTree` Node l v r

很容易（使用inorder的第一个定义）来编写

inTreeThenInorder : {x : a} -> (t : Tree a) -> x `InTree` t -> x `Elem` inorder t

结果有一个非常简单的结构，看起来相当不错。

编写

版本也不是非常困难

inorderThenInTree : x `Elem` inorder t -> x `InTree` t

不幸的是，到目前为止，我没有想出任何方法来编写inorderThenInTree的版本，我已经证明它是inTreeThenInorder的反转。我唯一提出的用途

listSplit : x `Elem` xs ++ ys -> Either (x `Elem` xs) (x `Elem` ys)

我在试图回到那里时遇到了麻烦。

我尝试了一些一般性的想法：

使用Vect代替List，可以更轻松地找出左侧的内容以及右侧的内容。我陷入了＆＃34;绿色粘液＆＃34;它的。
玩树旋转，尽可能地证明树根处的旋转导致了有根据的关系。（我没有在下面进行轮换，因为我从来没有找到一种方法来使用关于这些旋转的任何事情。）
尝试使用有关如何访问树节点的信息来装饰树节点。我没有花很长时间在这上面，因为我无法想出通过这种方法表达任何有趣的方法。
尝试构建我们在构建执行此功能的函数时返回原点的证明。这变得非常混乱，我被卡在某处或其他地方。

Answer 1

您使用inTreeThenInorder x (branch y l r) e with listSplit x (inOrder l) (y ∷ inOrder r) e引理走在了正确的轨道上。您可以使用该函数来了解目标元素是在树的左侧还是右侧。

这是我实施的相关行

.feature-box-main .feature-box:nth-child(1) {
    /*styles for first div here*/
}
.feature-box-main .feature-box:nth-child(2) {
    /*styles for second div here*/
}

这是完整的实施。我已将其作为外部链接包含在内，以避免不必要的剧透，并利用Agda精彩的HTML超链接，语法高亮输出。

http://www.galois.com/~emertens/agda-tree-inorder-elem/TreeElem.html

Answer 2

我在Idris写了inorderToFro和inorderFroTo以及相关的词条。 Here's the link.

关于您的解决方案我可以提出几点意见（没有详细介绍）：

首先，splitMiddle并非真的有必要。如果您对Right p = listSplit xs ys loc -> elemAppend xs ys p = loc使用更一般的splitRight类型，则可以涵盖相同的基础。

其次，您可以使用更多with模式而不是显式_lem函数;我认为它会更清晰，更简洁。

第三，你做了相当多的工作证明splitLeft和co。通常将函数的属性移动到函数中是有意义的。因此，我们可以修改listSplit以返回所需的证据，而不是单独编写listSplit及其结果的证明。这通常更容易实现。在我的解决方案中，我使用了以下类型：

data SplitRes : (x : a) -> (xs, ys : List a) -> (e : Elem x (xs ++ ys)) -> Type where
  SLeft  : (e' : Elem x xs) -> e' ++^ ys = e -> SplitRes x xs ys e
  SRight : (e' : Elem x ys) -> xs ^++ e' = e -> SplitRes x xs ys e

listSplit : (xs, ys : List a) -> (e : Elem x (xs ++ ys)) -> SplitRes x xs ys e

我本可以使用Either (e' : Elem x xs ** (e' ++^ ys = e)) (e' : Elem x ys ** (xs ^++ e' = e))代替SplitRes。但是，我遇到Either的问题。在我看来，伊德里斯的高阶统一太过摇摆不定;我无法理解为什么我的unsplitLeft函数不会与Either进行类型检查。 SplitRes在其类型中不包含函数，因此我猜这就是为什么它的工作更加顺畅。

一般来说，此时我推荐Agda而不是Idris来编写这样的样张。它检查得更快，而且更加强大和方便。我很惊讶你在这里写了这么多伊德里斯以及关于树遍历的另一个问题。

Answer 3

我能够弄清楚如何证明可以从树位置到列表位置，并从阅读glguy's answer中引用的引理类型返回。最终，我设法走了另一条路，虽然代码（下面）相当可怕。幸运的是，我能够重用可怕的列表引理来证明关于预订序遍历的相应定理。

module PreIn
import Data.List
%default total

data Tree : Type -> Type where
  Tip : Tree a
  Node : (l : Tree a) -> (v : a) -> (r : Tree a) -> Tree a
%name Tree t, u

data InTree : a -> Tree a -> Type where
  AtRoot : x `InTree` (Node l x r)
  OnLeft : x `InTree` l -> x `InTree` (Node l v r)
  OnRight : x `InTree` r -> x `InTree` (Node l v r)

onLeftInjective : OnLeft p = OnLeft q -> p = q
onLeftInjective Refl = Refl

onRightInjective : OnRight p = OnRight q -> p = q
onRightInjective Refl = Refl

noDups : Tree a -> Type
noDups t = (x : a) -> (here, there : x `InTree` t) -> here = there

noDupsList : List a -> Type
noDupsList xs = (x : a) -> (here, there : x `Elem` xs) -> here = there

inorder : Tree a -> List a
inorder Tip = []
inorder (Node l v r) = inorder l ++ [v] ++ inorder r

rotateInorder : (ll : Tree a) ->
                (vl : a) ->
                (rl : Tree a) ->
                (v : a) ->
                (r : Tree a) ->
                inorder (Node (Node ll vl rl) v r) = inorder (Node ll vl (Node rl v r))
rotateInorder ll vl rl v r =
   rewrite appendAssociative (vl :: inorder rl) [v] (inorder r)
   in rewrite sym $ appendAssociative (inorder rl) [v] (inorder r)
   in rewrite appendAssociative (inorder ll) (vl :: inorder rl) (v :: inorder r)
   in Refl


instance Uninhabited (Here = There y) where
  uninhabited Refl impossible

instance Uninhabited (x `InTree` Tip) where
  uninhabited AtRoot impossible

elemAppend : {x : a} -> (ys,xs : List a) -> x `Elem` xs -> x `Elem` (ys ++ xs)
elemAppend [] xs xInxs = xInxs
elemAppend (y :: ys) xs xInxs = There (elemAppend ys xs xInxs)

appendElem : {x : a} -> (xs,ys : List a) -> x `Elem` xs -> x `Elem` (xs ++ ys)
appendElem (x :: zs) ys Here = Here
appendElem (y :: zs) ys (There pr) = There (appendElem zs ys pr)

tThenInorder : {x : a} -> (t : Tree a) -> x `InTree` t -> x `Elem` inorder t
tThenInorder (Node l x r) AtRoot = elemAppend _ _ Here
tThenInorder (Node l v r) (OnLeft pr) = appendElem _ _ (tThenInorder _ pr)
tThenInorder (Node l v r) (OnRight pr) = elemAppend _ _ (There (tThenInorder _ pr))

listSplit_lem : (x,z : a) -> (xs,ys:List a) -> Either (x `Elem` xs) (x `Elem` ys)
  -> Either (x `Elem` (z :: xs)) (x `Elem` ys)
listSplit_lem x z xs ys (Left prf) = Left (There prf)
listSplit_lem x z xs ys (Right prf) = Right prf


listSplit : {x : a} -> (xs,ys : List a) -> x `Elem` (xs ++ ys) -> Either (x `Elem` xs) (x `Elem` ys)
listSplit [] ys xelem = Right xelem
listSplit (z :: xs) ys Here = Left Here
listSplit {x} (z :: xs) ys (There pr) = listSplit_lem x z xs ys (listSplit xs ys pr)

mutual
  inorderThenT : {x : a} -> (t : Tree a) -> x `Elem` inorder t -> InTree x t
  inorderThenT Tip xInL = absurd xInL
  inorderThenT {x} (Node l v r) xInL = inorderThenT_lem x l v r xInL (listSplit (inorder l) (v :: inorder r) xInL)

  inorderThenT_lem : (x : a) ->
                     (l : Tree a) -> (v : a) -> (r : Tree a) ->
                     x `Elem` inorder (Node l v r) ->
                     Either (x `Elem` inorder l) (x `Elem` (v :: inorder r)) ->
                     InTree x (Node l v r)
  inorderThenT_lem x l v r xInL (Left locl) = OnLeft (inorderThenT l locl)
  inorderThenT_lem x l x r xInL (Right Here) = AtRoot
  inorderThenT_lem x l v r xInL (Right (There locr)) = OnRight (inorderThenT r locr)

unsplitRight : {x : a} -> (e : x `Elem` ys) -> listSplit xs ys (elemAppend xs ys e) = Right e
unsplitRight {xs = []} e = Refl
unsplitRight {xs = (x :: xs)} e = rewrite unsplitRight {xs} e in Refl

unsplitLeft : {x : a} -> (e : x `Elem` xs) -> listSplit xs ys (appendElem xs ys e) = Left e
unsplitLeft {xs = []} Here impossible
unsplitLeft {xs = (x :: xs)} Here = Refl
unsplitLeft {xs = (x :: xs)} {ys} (There pr) =
  rewrite unsplitLeft {xs} {ys} pr in Refl

splitLeft_lem1 : (Left (There w) = listSplit_lem x y xs ys (listSplit xs ys z)) ->
                 (Left w = listSplit xs ys z) 

splitLeft_lem1 {w} {xs} {ys} {z} prf with (listSplit xs ys z)
  splitLeft_lem1 {w}  Refl | (Left w) = Refl
  splitLeft_lem1 {w}  Refl | (Right s) impossible

splitLeft_lem2 : Left Here = listSplit_lem x x xs ys (listSplit xs ys z) -> Void
splitLeft_lem2 {x} {xs} {ys} {z} prf with (listSplit xs ys z)
  splitLeft_lem2 {x = x} {xs = xs} {ys = ys} {z = z} Refl | (Left y) impossible
  splitLeft_lem2 {x = x} {xs = xs} {ys = ys} {z = z} Refl | (Right y) impossible

splitLeft : {x : a} -> (xs,ys : List a) ->
            (loc : x `Elem` (xs ++ ys)) ->
            Left e = listSplit {x} xs ys loc ->
            appendElem {x} xs ys e = loc
splitLeft {e} [] ys loc prf = absurd e
splitLeft (x :: xs) ys Here prf = rewrite leftInjective prf in Refl
splitLeft {e = Here} (x :: xs) ys (There z) prf = absurd (splitLeft_lem2 prf)
splitLeft {e = (There w)} (y :: xs) ys (There z) prf =
  cong $ splitLeft xs ys z (splitLeft_lem1 prf)

splitMiddle_lem3 : Right Here = listSplit_lem y x xs (y :: ys) (listSplit xs (y :: ys) z) ->
                   Right Here = listSplit xs (y :: ys) z

splitMiddle_lem3 {y} {x} {xs} {ys} {z} prf with (listSplit xs (y :: ys) z)
  splitMiddle_lem3 {y = y} {x = x} {xs = xs} {ys = ys} {z = z} Refl | (Left w) impossible
  splitMiddle_lem3 {y = y} {x = x} {xs = xs} {ys = ys} {z = z} prf | (Right w) =
    cong $ rightInjective prf  -- This funny dance strips the Rights off and then puts them
                               -- back on so as to change type.


splitMiddle_lem2 : Right Here = listSplit xs (y :: ys) pl ->
                   elemAppend xs (y :: ys) Here = pl

splitMiddle_lem2 {xs} {y} {ys} {pl} prf with (listSplit xs (y :: ys) pl) proof prpr
  splitMiddle_lem2 {xs = xs} {y = y} {ys = ys} {pl = pl} Refl | (Left loc) impossible
  splitMiddle_lem2 {xs = []} {y = y} {ys = ys} {pl = pl} Refl | (Right Here) = rightInjective prpr
  splitMiddle_lem2 {xs = (x :: xs)} {y = x} {ys = ys} {pl = Here} prf | (Right Here) = (\Refl impossible) prpr
  splitMiddle_lem2 {xs = (x :: xs)} {y = y} {ys = ys} {pl = (There z)} prf | (Right Here) =
    cong $ splitMiddle_lem2 {xs} {y} {ys} {pl = z} (splitMiddle_lem3 prpr)

splitMiddle_lem1 : Right Here = listSplit_lem y x xs (y :: ys) (listSplit xs (y :: ys) pl) ->
                   elemAppend xs (y :: ys) Here = pl

splitMiddle_lem1 {y} {x} {xs} {ys} {pl} prf with (listSplit xs (y :: ys) pl) proof prpr
  splitMiddle_lem1 {y = y} {x = x} {xs = xs} {ys = ys} {pl = pl} Refl | (Left z) impossible
  splitMiddle_lem1 {y = y} {x = x} {xs = xs} {ys = ys} {pl = pl} Refl | (Right Here) = splitMiddle_lem2 prpr

splitMiddle : Right Here = listSplit xs (y::ys) loc ->
              elemAppend xs (y::ys) Here = loc

splitMiddle {xs = []} prf = rightInjective prf
splitMiddle {xs = (x :: xs)} {loc = Here} Refl impossible
splitMiddle {xs = (x :: xs)} {loc = (There y)} prf = cong $ splitMiddle_lem1 prf

splitRight_lem1 : Right (There pl) = listSplit (q :: xs) (y :: ys) (There z) ->
                  Right (There pl) = listSplit xs (y :: ys) z

splitRight_lem1 {xs} {ys} {y} {z} prf with (listSplit xs (y :: ys) z)
  splitRight_lem1 {xs = xs} {ys = ys} {y = y} {z = z} Refl | (Left x) impossible
  splitRight_lem1 {xs = xs} {ys = ys} {y = y} {z = z} prf | (Right x) =
    cong $ rightInjective prf  -- Type dance: take the Right off and put it back on.

splitRight : Right (There pl) = listSplit xs (y :: ys) loc ->
             elemAppend xs (y :: ys) (There pl) = loc
splitRight {pl = pl} {xs = []} {y = y} {ys = ys} {loc = loc} prf = rightInjective prf
splitRight {pl = pl} {xs = (x :: xs)} {y = y} {ys = ys} {loc = Here} Refl impossible
splitRight {pl = pl} {xs = (x :: xs)} {y = y} {ys = ys} {loc = (There z)} prf =
  let rec = splitRight {pl} {xs} {y} {ys} {loc = z} in cong $ rec (splitRight_lem1 prf)


---------------------------
-- tThenInorder is a bijection from ways to find a particular element in a tree
-- and ways to find that element in its inorder traversal. `inorderToFro`
-- and `inorderFroTo` together demonstrate this by showing that `inorderThenT` is
-- its inverse.

||| `tThenInorder t` is a retraction of `inorderThenT t`
inorderFroTo : {x : a} -> (t : Tree a) -> (loc : x `Elem` inorder t) -> tThenInorder t (inorderThenT t loc) = loc
inorderFroTo Tip loc = absurd loc
inorderFroTo (Node l v r) loc with (listSplit (inorder l) (v :: inorder r) loc) proof prf
  inorderFroTo (Node l v r) loc | (Left here) =
    rewrite inorderFroTo l here in splitLeft _ _ loc prf
  inorderFroTo (Node l v r) loc | (Right Here) = splitMiddle prf
  inorderFroTo (Node l v r) loc | (Right (There x)) =
    rewrite inorderFroTo r x in splitRight prf

||| `inorderThenT t` is a retraction of `tThenInorder t`
inorderToFro : {x : a} -> (t : Tree a) -> (loc : x `InTree` t) -> inorderThenT t (tThenInorder t loc) = loc
inorderToFro (Node l v r) (OnLeft xInL) =
  rewrite unsplitLeft {ys = v :: inorder r} (tThenInorder l xInL)
  in cong $ inorderToFro _ xInL
inorderToFro (Node l x r) AtRoot =
  rewrite unsplitRight {x} {xs = inorder l} {ys = x :: inorder r} (tThenInorder (Node Tip x r) AtRoot)
  in Refl
inorderToFro {x} (Node l v r) (OnRight xInR) =
  rewrite unsplitRight {x} {xs = inorder l} {ys = v :: inorder r} (tThenInorder (Node Tip v r) (OnRight xInR))
  in cong $ inorderToFro _ xInR

如何在树和遍历之间建立双射？

3 个答案:

如何在树和遍历之间建立​​双射？

3 个答案:

如何在树和遍历之间建立双射？