include "basics/sums.ma".

include "cerco/BitVector.ma".

inductive BitVectorTrie (A: Type[0]): nat → Type[0] ≝
  Leaf: A → BitVectorTrie A O
| Node: ∀n: nat. BitVectorTrie A n → BitVectorTrie A n → BitVectorTrie A (S n)
| Stub: ∀n: nat. BitVectorTrie A n.

let rec lookup (A: Type[0]) (n: nat)
                (b: BitVector n) (t: BitVectorTrie A n) (a: A) on b
       : A ≝
  (match b return λx.λ_. x = n → A with
    [ VEmpty ⇒ 
      (match t return λx.λ_. O = x → A with
        [ Leaf l ⇒ λ_.l
        | Node h l r ⇒ λK.⊥
        | Stub s ⇒ λ_.a
        ])
    | VCons o hd tl ⇒
      match t return λx.λ_. (S o) = x → A with
        [ Leaf l ⇒ λK.⊥
        | Node h l r ⇒
           match hd with
             [ true ⇒ λK. lookup A h (tl⌈o ↦ h⌉) r a
             | false ⇒ λK. lookup A h (tl⌈o ↦ h⌉) l a
             ]
        | Stub s ⇒ λ_. a]
    ]) (refl ? n).
  [1,2:
    destruct
  |*:
    @ injective_S
    //
  ]
qed.

let rec prepare_trie_for_insertion (A: Type[0]) (n: nat) (b: BitVector n) (a:A) on b : BitVectorTrie A n ≝
   match b with
    [ VEmpty ⇒ Leaf A a
    | VCons o hd tl ⇒
      match hd with
        [ true ⇒  Node A o (Stub A o) (prepare_trie_for_insertion A o tl a)
        | false ⇒ Node A o (prepare_trie_for_insertion A o tl a) (Stub A o)
        ]
    ].

let rec insert (A: Type[0]) (n: nat) (b: BitVector n) (a: A) on b: BitVectorTrie A n → BitVectorTrie A n ≝
  (match b with
    [ VEmpty ⇒ λ_. Leaf A a
    | VCons o hd tl ⇒ λt.
          match t return λy.λ_. S o = y → BitVectorTrie A (S o) with
            [ Leaf l ⇒ λprf.⊥
            | Node p l r ⇒ λprf.
               match hd with
                [ true ⇒  Node A o (l⌈p ↦ o⌉) (insert A o tl a (r⌈p ↦ o⌉))
                | false ⇒ Node A o (insert A o tl a (l⌈p ↦ o⌉)) (r⌈p ↦ o⌉)
                ]
            | Stub p ⇒ λprf. (prepare_trie_for_insertion A ? (hd:::tl) a)
            ] (refl ? (S o))
    ]).
  [ destruct
  |*:
    @ injective_S
    //
  ]
qed.