在Agda中使用字符串作为键进行映射? [英] Map with Strings as Keys in Agda?

问题描述

我在弄清楚如何正确地在Agda中使用String键制作Map时遇到了一些麻烦.我有以下内容:

I'm having some trouble figuring out how to properly make a Map with String keys in Agda. I've got the following:

import Data.AVL.IndexedMap

Var = String

data Type where -- ...

alwaysType : Var -> Set
alwaysType _ = Type

open Data.AVL.IndexedMap alwaysType (StrictTotalOrder.isStrictTotalOrder Data.String.strictTotalOrder) 

这给出了错误:

String != Σ String _Key_90 of type Set
when checking that the expression
StrictTotalOrder.isStrictTotalOrder strictTotalOrder has type
Relation.Binary.IsStrictTotalOrder .Agda.Builtin.Equality._≡_
__<__91

打开地图"模块的正确方法是什么?

What is the proper way to open the Map module?

推荐答案

Data.AVL.IndexedMap模块用于(有限)映射，其中存在键和值的类型族，以及与给定键关联的值与值共享索引.

The Data.AVL.IndexedMap module is for (finite) maps where there is a family of types for the keys and the values, and the value associated with a given key shares the index with the value.

这不是您想要的，因为您希望所有键都为String s.因此，只需使用Data.AVL(即具有未索引键的版本):

This is not what you want here, since you want all your keys to be Strings. So just use Data.AVL (i.e. the version with non-indexed keys):

open import Data.String using (String)
open import Function using (const)

Key = String

postulate
  Value : Set

open import Relation.Binary using (StrictTotalOrder)
open import Data.AVL (const Value) (StrictTotalOrder.isStrictTotalOrder Data.String.strictTotalOrder)

不幸的是，这仍然没有进行类型检查:

Unfortunately, this still doesn't typecheck:

.Relation.Binary.List.Pointwise.Rel
(StrictTotalOrder._≈_ .Data.Char.strictTotalOrder)
(Data.String.toList x) (Data.String.toList x₁)
!= x .Agda.Builtin.Equality.≡ x₁ of type Set
when checking that the expression
StrictTotalOrder.isStrictTotalOrder Data.String.strictTotalOrder
has type IsStrictTotalOrder .Agda.Builtin.Equality._≡_ __<__10

这是因为Data.String.strictTotalOrder使用逐点相等(在组成String的Char的Char的ℕ值的列表上)，并且Data.AVL需要命题相等.因此，完全相同的示例将适用于例如ℕ键:

That's because Data.String.strictTotalOrder uses pointwise equality (over the list of ℕ values of the Chars that make up the String), and Data.AVL requires propositional equality. So the exact same example would work with, e.g., ℕ keys:

open import Data.Nat using (ℕ)
open import Function using (const)

Key = ℕ

postulate
  Value : Set

import Data.Nat.Properties
open import Relation.Binary using (StrictTotalOrder)

open import Data.AVL (const Value) (StrictTotalOrder.isStrictTotalOrder Data.Nat.Properties.strictTotalOrder)

因此下一步需要将StrictTotalOrder.isStrictTotalOrder Data.String.strictTotalOrder转换为IsStrictTotalOrder (_≡_ {A = String}) _类型的东西.我现在暂时将其留给其他人，但是如果有时间，如果您无法正常工作并且也没有其他人来找它，我很乐意稍后再进行研究.

So the next step needs to be to transform StrictTotalOrder.isStrictTotalOrder Data.String.strictTotalOrder into something of type IsStrictTotalOrder (_≡_ {A = String}) _. I'll leave that to someone else for now, but I'm happy to look into it later, when I have the time, if you can't get it working and noone else picks it up either.

编辑后添加:这是一种(可能非常复杂)将标准库中的StrictTotalOrder String转换为使用命题相等的方法:

EDITED TO ADD: Here's a (possibly horribly over-complicated) way of turning that StrictTotalOrder for Strings from the standard lib into something that uses propositional equality:

open import Function using (const; _∘_; _on_)
open import Relation.Binary

open import Data.String
  using (String; toList∘fromList; fromList∘toList)
  renaming (toList to stringToList; fromList to stringFromList)

open import Relation.Binary.List.Pointwise as Pointwise
open import Relation.Binary.PropositionalEquality as P hiding (trans)
open import Data.Char.Base renaming (toNat to charToNat)

STO : StrictTotalOrder _ _ _
STO = record
  { Carrier = String
  ; _≈_ = _≡_
  ; _<_ = _<_
  ; isStrictTotalOrder = record
    { isEquivalence = P.isEquivalence
    ; trans = λ {x} {y} {z} → trans {x} {y} {z}
    ; compare = compare
    }
  }
  where
    open StrictTotalOrder Data.String.strictTotalOrder 
      renaming (isEquivalence to string-isEquivalence; compare to string-compare)

    -- It feels like this should be defined somewhere in the
    -- standard library, but I can't find it...
    primCharToNat-inj : ∀ {x y} → primCharToNat x ≡ primCharToNat y → x ≡ y
    primCharToNat-inj _ = trustMe
      where
        open import Relation.Binary.PropositionalEquality.TrustMe

    open import Data.List
    lem : ∀ {xs ys} → Pointwise.Rel (_≡_ on primCharToNat) xs ys → xs ≡ ys
    lem [] = P.refl
    lem {x ∷ xs} {y ∷ ys} (x∼y ∷ p) with primCharToNat-inj {x} {y} x∼y
    lem {x ∷ xs} {_ ∷ ys} (x∼y ∷ p) | P.refl = cong _ (lem p)

    ≡-from-≈ : {s s′ : String} → s ≈ s′ → s ≡ s′
    ≡-from-≈ {s} {s′} p = begin
         s ≡⟨ sym (fromList∘toList _) ⟩
         stringFromList (stringToList s) ≡⟨ cong stringFromList (lem p) ⟩
         stringFromList (stringToList s′) ≡⟨ fromList∘toList _ ⟩
         s′ ∎
      where
        open P.≡-Reasoning

    ≈-from-≡ : {s s′ : String} → s ≡ s′ → s ≈ s′
    ≈-from-≡ {s} {_} refl = string-refl {s}
      where
        open IsEquivalence string-isEquivalence renaming (refl to string-refl) using ()

    compare : (x y : String) → Tri (x < y) (x ≡ y) _
    compare x y with string-compare x y
    compare x y | tri< a ¬b ¬c = tri< a (¬b ∘ ≈-from-≡) ¬c
    compare x y | tri≈ ¬a b ¬c = tri≈ ¬a (≡-from-≈ b) ¬c
    compare x y | tri> ¬a ¬b c = tri> ¬a (¬b ∘ ≈-from-≡) c

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