由类型索引 vs 包含 idris 中的类型

Question

我目前正在阅读Type-Driven Development with Idris 的书。

关于第 6 章中示例数据存储的设计，我有两个问题。数据存储是一个命令行应用程序，允许用户设置其中存储的数据类型，然后添加新数据。

以下是代码的相关部分（略作简化）。你可以在 Github 上看到full code：

module Main

import Data.Vect

infixr 4 .+.

-- This datatype is to define what sorts of data can be contained in the data store.
data Schema
  = SString
  | SInt
  | (.+.) Schema Schema

-- This is a type-level function that translates a Schema to an actual type.
SchemaType : Schema -> Type
SchemaType SString = String
SchemaType SInt = Int
SchemaType (x .+. y) = (SchemaType x, SchemaType y)

-- This is the data store.  It can potentially be storing any sort of schema.
record DataStore where
       constructor MkData
       schema : Schema
       size : Nat
       items : Vect size (SchemaType schema)

-- This adds new data to the datastore, making sure that the new data is
-- the same type that the DataStore holds.
addToStore
  : (dataStore : DataStore) -> SchemaType (schema dataStore) -> DataStore
addToStore (MkData schema' size' store') newitem =
  MkData schema' _ (addToData store')
  where
    addToData
      : Vect size' (SchemaType schema') -> Vect (size' + 1) (SchemaType schema')
    addToData xs = xs ++ [newitem]

-- These are commands the user can use on the command line.  They are able
-- to change the schema, and add new data.
data Command : Schema -> Type where
  SetSchema : (newSchema : Schema) -> Command schema
  Add : SchemaType schema -> Command schema

-- Given a Schema, this parses input from the user into a Command.
parse : (schema : Schema) -> String -> Maybe (Command schema)

-- This is the main workhorse of the application.  It parses user
-- input, turns it into a command, then evaluates the command and 
-- returns an updated DataStore.
processInput
  : (dataStore : DataStore) -> String -> Maybe (String, DataStore)
processInput dataStore@(MkData schema' size' items') input =
  case parse schema' input of
    Nothing => Just ("Invalid command\n", dataStore)
    Just (SetSchema newSchema) =>
      Just ("updated schema and reset datastore\n", MkData newSchema _ [])
    Just (Add item) =>
      let newStore = addToStore (MkData schema' size' items') item
      in Just ("ID " ++ show (size dataStore) ++ "\n", newStore)

-- This kicks off processInput with an empty datastore and a simple Schema
-- of SString.
main : IO ()
main = replWith (MkData SString _ []) "Command: " processInput

这很有意义且易于使用，但有一点让我对设计感到烦恼。 为什么DataStore 包含Schema 而不是被一个索引？

因为DataStore 没有被Schema 索引，我们可能编写了一个错误的addToStore 函数，如下所示：

addToStore
  : (dataStore : DataStore) -> SchemaType (schema dataStore) -> DataStore
addToStore _ newitem =
  MkData SInt _ []

这是我想象的更多类型安全代码的样子。你可以在 Github 上看到full code：

module Main

import Data.Vect

infixr 4 .+.

data Schema
  = SString
 | SInt
 | (.+.) Schema Schema

SchemaType : Schema -> Type
SchemaType SString = String
SchemaType SInt = Int
SchemaType (x .+. y) = (SchemaType x, SchemaType y)

record DataStore (schema : Schema) where
       constructor MkData
       size : Nat
       items : Vect size (SchemaType schema)

addToStore
  : (dataStore : DataStore schema) ->
    SchemaType schema ->
    DataStore schema
addToStore {schema} (MkData size' store') newitem =
  MkData _ (addToData store')
  where
    addToData
      : Vect size' (SchemaType schema) -> Vect (size' + 1) (SchemaType schema)
    addToData xs = xs ++ [newitem]

data Command : Schema -> Type where
  SetSchema : (newSchema : Schema) -> Command schema
  Add : SchemaType schema -> Command schema

parse : (schema : Schema) -> String -> Maybe (Command schema)

processInput
  : (schema : Schema ** DataStore schema) ->
    String ->
    Maybe (String, (newschema ** DataStore newschema))
processInput (schema ** (MkData size' items')) input =
  case parse schema input of
    Nothing => Just ("Invalid command\n", (_ ** MkData size' items'))
    Just (SetSchema newSchema) =>
      Just ("updated schema and reset datastore\n", (newSchema ** MkData _ []))
    Just (Add item) =>
      let newStore = addToStore (MkData size' items') item
          msg = "ID " ++ show (size newStore) ++ "\n"
      in Just (msg, (schema ** newStore))

main : IO ()
main = replWith (SString ** MkData _ []) "Command: " processInput

---

这是我的两个问题：

1. 为什么这本书没有使用DataStore 类型的更安全的版本（Schema 索引的那个）？使用不太安全的版本（只包含Schema）有什么好处吗？

2. 被另一种类型索引的类型与包含另一种类型的理论区别是什么？这种差异会根据您使用的语言而改变吗？

   例如，我想 Idris 可能没有太大的不同，但 Haskell 有很大的不同。 （对吧……？）

   我刚开始玩 Idris（而且我对在 Haskell 中使用单例或 GADT 不是特别精通），所以我很难理解这一点。如果您能指出任何有关这方面的论文，我会非常感兴趣。

Answer 1

根据 cmets，这是迂腐的。早期，使用依赖记录，因此不需要处理类型索引。后来，使用索引类型来限制（并通过证明搜索更容易找到）有效实现。