source: src/ASM/BitVectorTrie.ma @ 990

include "basics/types.ma".

include "utilities/option.ma".
include "ASM/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.

axiom fold:
 ∀A, B: Type[0].
 ∀n: nat.
 ∀f: BitVector n → A → B → B.
 ∀t: BitVectorTrie A n.
 ∀b: B.
   B.

let rec lookup_opt (A: Type[0]) (n: nat)
                (b: BitVector n) (t: BitVectorTrie A n) on t
       : option A ≝
 (match t return λx.λ_. BitVector x → option A with
  [ Leaf l ⇒ λ_.Some ? l
  | Node h l r ⇒ λb. lookup_opt A ? (tail … b) (if head' … b then r else l)
  | Stub _ ⇒ λ_.None ?
  ]) b.

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.

let rec update (A: Type[0]) (n: nat) (b: BitVector n) (a: A) on b: BitVectorTrie A n → option (BitVectorTrie A n) ≝
  (match b with
    [ VEmpty ⇒ λt. match t return λy.λ_. O = y → option (BitVectorTrie A O) with
                   [ Leaf _ ⇒ λ_. Some ? (Leaf A a)
                   | Stub _ ⇒ λ_. None ?
                   | Node _ _ _ ⇒ λprf. ⊥
                   ] (refl ? O)
    | VCons o hd tl ⇒ λt.
          match t return λy.λ_. S o = y → option (BitVectorTrie A (S o)) with
            [ Leaf l ⇒ λprf.⊥
            | Node p l r ⇒ λprf.
               match hd with
                [ true ⇒  option_map ?? (λv. Node A o (l⌈p ↦ o⌉) v) (update A o tl a (r⌈p ↦ o⌉))
                | false ⇒ option_map ?? (λv. Node A o v (r⌈p ↦ o⌉)) (update A o tl a (l⌈p ↦ o⌉))
                ]
            | Stub p ⇒ λprf. None ?
            ] (refl ? (S o))
    ]).
  [ 1,2: destruct
  |*:
    @ injective_S @sym_eq @prf
  ]
qed.

let rec merge (A: Type[0]) (n: nat) (b: BitVectorTrie A n) on b: BitVectorTrie A n → BitVectorTrie A n ≝
  match b return λx. λ_. BitVectorTrie A x → BitVectorTrie A x with
  [ Stub _ ⇒ λc. c
  | Leaf l ⇒ λc. match c with [ Leaf a ⇒ Leaf ? a | _ ⇒ Leaf ? l ]
  | Node p l r ⇒
    λc.
    (match c return λx. λ_. x = (S p) → BitVectorTrie A (S p) with
    [ Node p' l' r' ⇒ λprf. Node ? ? (merge ?? l (l'⌈p' ↦ p⌉)) (merge ?? r (r'⌈p' ↦ p⌉))
    | Stub _ ⇒ λprf. Node ? p l r
    | Leaf _ ⇒ λabsd. ?
    ] (refl ? (S p)))
  ].
  [1:
      destruct(absd)
  |2,3:
      @ injective_S
        assumption
  ]
qed.

lemma BitVectorTrie_O:
 ∀A:Type[0].∀v:BitVectorTrie A 0.(∃w. v ≃ Leaf A w) ∨ v ≃ Stub A 0.
 #A #v generalize in match (refl … O) cases v in ⊢ (??%? → (?(??(λ_.?%%??)))(?%%??))
  [ #w #_ %1 %[@w] %
  | #n #l #r #abs @⊥ destruct(abs)
  | #n #EQ %2 >EQ %]
qed.

lemma BitVectorTrie_Sn:
 ∀A:Type[0].∀n.∀v:BitVectorTrie A (S n).(∃l,r. v ≃ Node A n l r) ∨ v ≃ Stub A (S n).
 #A #n #v generalize in match (refl … (S n)) cases v in ⊢ (??%? → (?(??(λ_.??(λ_.?%%??))))%)
  [ #m #abs @⊥ destruct(abs)
  | #m #l #r #EQ %1 <(injective_S … EQ) %[@l] %[@r] //
  | #m #EQ %2 // ]
qed.

lemma lookup_prepare_trie_for_insertion_hit:
 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.
  lookup … b (prepare_trie_for_insertion … b v) a = v. 
 #A #a #v #n #b elim b // #m #hd #tl #IH cases hd normalize //
qed.
 
lemma lookup_insert_hit:
 ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n.
  lookup … b (insert … b v t) a = v.
 #A #a #v #n #b elim b -b -n //
 #n #hd #tl #IH #t cases(BitVectorTrie_Sn … t)
  [ * #l * #r #JMEQ >JMEQ cases hd normalize //
  | #JMEQ >JMEQ cases hd normalize @lookup_prepare_trie_for_insertion_hit ]
qed.

lemma lookup_prepare_trie_for_insertion_miss:
 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.
  (notb (eq_bv ? b c)) → lookup … b (prepare_trie_for_insertion … c v) a = a.
 #A #a #v #n #c elim c
  [ #b >(BitVector_O … b) normalize #abs @⊥ //
  | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
    cases hd cases hd' normalize
    [2,3: #_ cases tl' //
    |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) /2/ ]]
qed.
  
lemma lookup_insert_miss:
 ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n.
  (notb (eq_bv ? b c)) → lookup … b (insert … c v t) a = lookup … b t a.
 #A #a #v #n #c elim c -c -n
  [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF //
  | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ
    #t cases(BitVectorTrie_Sn … t)
    [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize // @IH //
    | #JMEQ >JMEQ cases hd cases hd' #H normalize in H;
     [1,4: change in H with (bool_to_Prop (notb (eq_bv ???))) ] normalize
     [3,4: cases tl' // | *: @lookup_prepare_trie_for_insertion_miss //]]]
qed.