```------------------------------------------------------------------------
-- The Agda standard library
--
-- Non empty trie, basic type and operations
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --sized-types #-}

open import Relation.Binary using (StrictTotalOrder)

module Data.Trie.NonEmpty {k e r} (S : StrictTotalOrder k e r) where

open import Level
open import Size
open import Data.Product as Prod using (∃; uncurry; -,_)
open import Data.List.Base as List using (List; []; _∷_; _++_)
open import Data.List.NonEmpty as List⁺ using (List⁺; [_]; concatMap)
open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′) hiding (module Maybe)
open import Data.These as These using (These; this; that; these)
open import Function as F
import Function.Identity.Effectful as Identity
open import Relation.Unary using (_⇒_; IUniversal)

open StrictTotalOrder S
using (module Eq)
renaming (Carrier to Key)

open import Data.List.Relation.Binary.Equality.Setoid Eq.setoid
open import Data.Tree.AVL.Value hiding (Value)
open import Data.Tree.AVL.Value ≋-setoid using (Value)
open import Data.Tree.AVL.NonEmpty S as Tree⁺ using (Tree⁺)
open Value

------------------------------------------------------------------------
-- Definition

-- A Trie⁺ is a tree branching over an alphabet of Keys. It stores values
-- indexed over the Word (i.e. List Key) that was read to reach them.
-- Each node in the Trie⁺ contains either a value, a non-empty Tree of
-- sub-Trie⁺ reached by reading an extra letter, or both.

Word : Set k
Word = List Key

eat : ∀ {v} → Value v → Word → Value v
family   (eat V ks) = family   V ∘′ (ks ++_)
respects (eat V ks) = respects V ∘′ ++⁺ ≋-refl

data Trie⁺ {v} (V : Value v) : Size → Set (v ⊔ k ⊔ e ⊔ r)
Tries⁺ : ∀ {v} (V : Value v) (i : Size) → Set (v ⊔ k ⊔ e ⊔ r)

map : ∀ {v w} (V : Value v) (W : Value w) {i} →
∀[ family V ⇒ family W ] →
Trie⁺ V i → Trie⁺ W i

data Trie⁺ V where
node : ∀ {i} → These (family V []) (Tries⁺ V i) → Trie⁺ V (↑ i)

Tries⁺ V j = Tree⁺ \$ MkValue (λ k → Trie⁺ (eat V (k ∷ [])) j)
\$ λ eq → map _ _ (respects V (eq ∷ ≋-refl))

map V W f (node t) = node \$ These.map f (Tree⁺.map (map _ _ f)) t

------------------------------------------------------------------------
-- Query

lookup : ∀ {v} {V : Value v} → Trie⁺ V ∞ → ∀ ks →
Maybe (These (family V ks) (Tries⁺ (eat V ks) ∞))
lookup (node nd) []       = just (These.map₂ (Tree⁺.map id) nd)
lookup (node nd) (k ∷ ks) = let open Maybe in do
ts ← These.fromThat nd
t  ← Tree⁺.lookup ts k
lookup t ks

module _ {v} {V : Value v} where

lookupValue : Trie⁺ V ∞ → (ks : Word) → Maybe (family V ks)
lookupValue t ks = lookup t ks Maybe.>>= These.fromThis

lookupTries⁺ : Trie⁺ V ∞ → ∀ ks → Maybe (Tries⁺ (eat V ks) ∞)
lookupTries⁺ t ks = lookup t ks Maybe.>>= These.fromThat

lookupTrie⁺ : Trie⁺ V ∞ → ∀ k → Maybe (Trie⁺ (eat V (k ∷ [])) ∞)
lookupTrie⁺ t k = lookupTries⁺ t [] Maybe.>>= λ ts → Tree⁺.lookup ts k

------------------------------------------------------------------------
-- Construction

singleton : ∀ {v} {V : Value v} ks → family V ks → Trie⁺ V ∞
singleton []       v = node (this v)
singleton (k ∷ ks) v = node (that (Tree⁺.singleton k (singleton ks v)))

insertWith : ∀ {v} {V : Value v} ks → (Maybe (family V ks) → family V ks) →
Trie⁺ V ∞ → Trie⁺ V ∞
insertWith []       f (node nd) =
node (These.fold (this ∘ f ∘ just) (these (f nothing)) (these ∘ f ∘ just) nd)
insertWith {v} {V} (k ∷ ks) f (node {j} nd) = node \$
These.fold (λ v → these v (Tree⁺.singleton k end))
(that ∘′ rec)
(λ v → these v ∘′ rec)
nd
where

end : Trie⁺ (eat V (k ∷ [])) ∞
end = singleton ks (f nothing)

rec : Tries⁺ V ∞ → Tries⁺ V ∞
rec = Tree⁺.insertWith k (maybe′ (insertWith ks f) end)

module _ {v} {V : Value v} where

private Val = family V

insert : ∀ ks → Val ks → Trie⁺ V ∞ → Trie⁺ V ∞
insert ks = insertWith ks ∘′ F.const

fromList⁺ : List⁺ (∃ Val) → Trie⁺ V ∞
fromList⁺ = List⁺.foldr (uncurry insert) (uncurry singleton)

toList⁺ : ∀ {v} {V : Value v} {i} → let Val = Value.family V in
Trie⁺ V i → List⁺ (∃ Val)
toList⁺ (node nd) = These.mergeThese List⁺._⁺++⁺_
\$ These.map fromVal fromTries⁺ nd
where

fromVal    = [_] ∘′ -,_
fromTrie⁺  = λ (k , v) → List⁺.map (Prod.map (k ∷_) id) (toList⁺ v)
fromTries⁺ = concatMap fromTrie⁺ ∘′ Tree⁺.toList⁺

------------------------------------------------------------------------
-- Modification

-- Deletion

deleteWith : ∀ {v} {V : Value v} {i} ks →
(∀ {i} → Trie⁺ (eat V ks) i → Maybe (Trie⁺ (eat V ks) i)) →
Trie⁺ V i → Maybe (Trie⁺ V i)
deleteWith []       f t           = f t
deleteWith (k ∷ ks) f t@(node nd) = let open RawMonad Identity.monad in do
just ts ← These.fromThat nd where _ → just t
-- This would be a lot cleaner if we had
-- Tree⁺.updateWith : ∀ k → (Maybe (V k) → Maybe (V k)) → AVL → AVL
-- Instead we lookup the subtree, update it and either put it back in
-- or delete the corresponding leaf depending on whether the result is successful.
just t′ ← Tree⁺.lookup ts k where _ → just t
Maybe.map node ∘′ Maybe.align (These.fromThis nd) \$′ case deleteWith ks f t′ of λ where
nothing  → Tree⁺.delete k ts
(just u) → just (Tree⁺.insert k u ts)

module _ {v} {V : Value v} where

deleteTrie⁺ : ∀ {i} (ks : Word) → Trie⁺ V i → Maybe (Trie⁺ V i)
deleteTrie⁺ ks = deleteWith ks (F.const nothing)

deleteValue : ∀ {i} (ks : Word) → Trie⁺ V i → Maybe (Trie⁺ V i)
deleteValue ks = deleteWith ks \$ λ where
(node nd) → Maybe.map node \$′ These.deleteThis nd

deleteTries⁺ : ∀ {i} (ks : Word) → Trie⁺ V i → Maybe (Trie⁺ V i)
deleteTries⁺ ks = deleteWith ks \$ λ where
(node nd) → Maybe.map node \$′ These.deleteThat nd
```