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(* BitVector.ma: Fixed length bitvectors, and common operations on them.      *)
(*               Most functions are specialised versions of those found in    *)
(*               Vector.ma as a courtesy, or Boolean functions lifted into    *)
(*               BitVector variants.                                          *)
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include "Universes.ma".

include "Vector.ma".
include "List.ma".
include "Nat.ma".
include "Bool.ma".

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(* Common synonyms.                                                           *)
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ndefinition BitVector ≝ λn: Nat. Vector Bool n.
ndefinition Bit ≝ BitVector (S Z).
ndefinition Nibble ≝ BitVector (S (S (S (S Z)))).
ndefinition Byte7 ≝ BitVector (S (S (S (S (S (S (S Z))))))).
ndefinition Byte ≝ BitVector (S (S (S (S (S (S (S (S Z)))))))).
ndefinition Word ≝ BitVector (S (S (S (S (S (S (S (S (S (S (S (S (S (S (S (S Z)))))))))))))))).
ndefinition Word11 ≝ BitVector (S (S (S (S (S (S (S (S (S (S (S (S (S Z))))))))))))).

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(* Lookup.                                                                    *)
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ndefinition get_index ≝
  λn: Nat.
  λb: BitVector n.
  λm: Nat.
    get_index Bool n b m.
    
interpretation "BitVector get_index" 'get_index b n = (get_index ? b n).
    
ndefinition set_index ≝
  λn: Nat.
  λb: BitVector n.
  λm: Nat.
    set_index Bool n b m.
    
ndefinition drop ≝
  λn: Nat.
  λb: BitVector n.
  λm: Nat.
    drop Bool n b m.

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(* Creating bitvectors from scratch.                                          *)
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ndefinition zero ≝
  λn: Nat.
    replicate Bool n False.
    
ndefinition max ≝
  λn: Nat.
    replicate Bool n True.
    
ndefinition append ≝
  λm, n: Nat.
  λb: BitVector m.
  λc: BitVector n.
    append Bool m n b c.
    
interpretation "BitVector append" 'append b c = (append b c).

ndefinition pad ≝
  λm, n: Nat.
  λb: BitVector n.
    let padding ≝ replicate Bool m False in
      append Bool m n padding b.
      
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(* Other manipulations.                                                       *)
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ndefinition reverse ≝
  λn: Nat.
  λb: BitVector n.
    reverse Bool n b.

ndefinition length ≝
  λn: Nat.
  λb: BitVector n.
    length Bool n b.

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(* Logical operations.                                                        *)
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ndefinition conjunction ≝
  λn: Nat.
  λb: BitVector n.
  λc: BitVector n.
    zip_with Bool Bool Bool n conjunction b c.
    
interpretation "BitVector conjunction" 'conjunction b c = (conjunction ? b c).
   
ndefinition inclusive_disjunction ≝
  λn: Nat.
  λb: BitVector n.
  λc: BitVector n.
    zip_with Bool Bool Bool n inclusive_disjunction b c.
    
interpretation "BitVector inclusive disjunction"
  'inclusive_disjunction b c = (inclusive_disjunction ? b c).
         
ndefinition exclusive_disjunction ≝
  λn: Nat.
  λb: BitVector n.
  λc: BitVector n.
    zip_with Bool Bool Bool n exclusive_disjunction b c.
    
interpretation "BitVector exclusive disjunction"
  'exclusive_disjunction b c = (exclusive_disjunction ? b c).
    
ndefinition negation ≝
  λn: Nat.
  λb: BitVector n.
    map Bool Bool n (negation) b.
    
interpretation "BitVector negation" 'negation b c = (negation ? b c).

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(* Rotates and shifts.                                                        *)
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ndefinition rotate_left ≝
  λm, n: Nat.
  λb: BitVector n.
    rotate_left Bool n m b.

ndefinition rotate_right ≝
  λm, n: Nat.
  λb: BitVector n.
    rotate_right Bool n m b.
    
ndefinition shift_left ≝
  λm, n: Nat.
  λb: BitVector n.
  λc: Bool.
    shift_left Bool n m b c.
    
ndefinition shift_right ≝
  λm, n: Nat.
  λb: BitVector n.
  λc: Bool.
    shift_right Bool n m b c.
    
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(* Conversions to and from lists and natural numbers.                         *)
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ndefinition list_of_bitvector ≝
  λn: Nat.
  λb: BitVector n.
    list_of_vector Bool n b.
    
ndefinition bitvector_of_list ≝
  λl: List Bool.
    vector_of_list Bool l.

nlet rec bitvector_of_nat_aux (n: Nat) on n ≝
  let divrem ≝ divide_with_remainder n (S (S Z)) in
  let div ≝ first Nat Nat divrem in
  let rem ≝ second Nat Nat divrem in
    match div with
      [ Z ⇒
        match rem with
          [ Z ⇒ Empty Bool
          | S r ⇒ ? (True :: (bitvector_of_nat_aux Z))
          ]
      | S d ⇒
        match rem with
          [ Z ⇒ ? (False :: (bitvector_of_nat_aux (S d)))
          | S r ⇒ ? (True :: (bitvector_of_nat_aux (S d)))
          ]
      ].
      //.
nqed.

ndefinition bitvector_of_nat ≝
  λm, n: Nat.
    let biglist ≝ reverse Bool (bitvector_of_nat_aux m) in
    let size ≝ length Bool biglist in
    let bitvector ≝ bitvector_of_list biglist in
    let difference ≝ n - size in
      pad difference size bitvector.

nlet rec nat_of_bitvector (n: Nat) (b: BitVector n) on b ≝
  match b with
    [ Empty ⇒ Z
    | Cons o hd tl ⇒
      let hdval ≝ match hd with [ True ⇒ S Z | False ⇒ Z ] in
        ((exponential (S (S Z)) o) * hdval) + nat_of_bitvector o tl
    ].