Package word-bits: Word to bit-list conversions

Information

Files

• Package tarball word-bits-1.100.tgz
• Theory source file word-bits.thy (included in the package tarball)

Defined Constants

• Data
  • Word
    • and
    • bit
    • not
    • or
    • shiftLeft
    • shiftRight
    • Bits
      • compare
      • fromWord
      • normal
      • toWord

Theorems

⊦ ∀w. normal (fromWord w)

⊦ toWord [] = 0

⊦ toNatural (toWord []) = 0

⊦ ∀w. toWord (fromWord w) = w

⊦ ∀w. length (fromWord w) = width

⊦ ∀w. Bits.width (toNatural w) ≤ width

⊦ ∀q. compare q [] [] ⇔ q

⊦ ∀w. Bits.fromList (fromWord w) = toNatural w

⊦ ∀l. toWord l = fromNatural (Bits.fromList l)

⊦ ∀w. fromWord w = Bits.toVector (toNatural w) width

⊦ ∀w. Bits.bound (toNatural w) width = toNatural w

⊦ ∀n. toNatural (fromNatural n) = Bits.bound n width

⊦ ∀l. normal l ⇔ length l = width

⊦ ∀w. not w = toWord (map (¬) (fromWord w))

⊦ ∀l. normal l ⇔ fromWord (toWord l) = l

⊦ ∀w n. bit w n ⇔ Bits.bit (toNatural w) n

⊦ ∀l. toNatural (toWord l) < 2 ↑ length l

⊦ ∀w₁ w₂. fromWord w₁ = fromWord w₂ ⇔ w₁ = w₂

⊦ ∀w₁ w₂. fromWord w₁ = fromWord w₂ ⇒ w₁ = w₂

⊦ ∀w n. shiftLeft w n = fromNatural (Bits.shiftLeft (toNatural w) n)

⊦ ∀w n. shiftRight w n = fromNatural (Bits.shiftRight (toNatural w) n)

⊦ ∀w₁ w₂. and w₁ w₂ = fromNatural (Bits.and (toNatural w₁) (toNatural w₂))

⊦ ∀w₁ w₂. or w₁ w₂ = fromNatural (Bits.or (toNatural w₁) (toNatural w₂))

⊦ ∀w₁ w₂. compare ⊥ (fromWord w₁) (fromWord w₂) ⇔ w₁ < w₂

⊦ ∀w₁ w₂. compare ⊤ (fromWord w₁) (fromWord w₂) ⇔ w₁ ≤ w₂

⊦ ∀w₁ w₂. and w₁ w₂ = toWord (zipWith (∧) (fromWord w₁) (fromWord w₂))

⊦ ∀w₁ w₂. or w₁ w₂ = toWord (zipWith (∨) (fromWord w₁) (fromWord w₂))

⊦ ∀l n. shiftLeft (toWord l) n = toWord (replicate ⊥ n @ l)

⊦ ∀k w. bit (not w) k ⇔ k < width ∧ ¬bit w k

⊦ ∀k w₁ w₂. bit (and w₁ w₂) k ⇔ bit w₁ k ∧ bit w₂ k

⊦ ∀k w₁ w₂. bit (or w₁ w₂) k ⇔ bit w₁ k ∨ bit w₂ k

⊦ ∀q w₁ w₂.
    compare q (fromWord w₁) (fromWord w₂) ⇔ if q then w₁ ≤ w₂ else w₁ < w₂

⊦ ∀w₁ w₂. (∀i. i < width ⇒ (bit w₁ i ⇔ bit w₂ i)) ⇒ w₁ = w₂

⊦ ∀l n. bit (toWord l) n ⇔ n < width ∧ n < length l ∧ nth l n

⊦ ∀q l₁ l₂.
    length l₁ = length l₂ ⇒
    (compare q l₁ l₂ ⇔
     Bits.compare q (Bits.fromList l₁) (Bits.fromList l₂))

⊦ ∀l.
    fromWord (toWord l) =
    if length l ≤ width then l @ replicate ⊥ (width - length l)
    else take width l

⊦ ∀h t.
    toWord (h :: t) =
    if h then 1 + shiftLeft (toWord t) 1 else shiftLeft (toWord t) 1

⊦ ∀l n.
    length l ≤ width ⇒
    shiftRight (toWord l) n =
    if length l ≤ n then toWord [] else toWord (drop n l)

⊦ ∀l n.
    width ≤ length l ⇒
    shiftRight (toWord l) n =
    if width ≤ n then toWord [] else toWord (drop n (take width l))

⊦ ∀q h₁ h₂ t₁ t₂.
    compare q (h₁ :: t₁) (h₂ :: t₂) ⇔
    compare (¬h₁ ∧ h₂ ∨ ¬(h₁ ∧ ¬h₂) ∧ q) t₁ t₂

⊦ ∀l n.
    shiftRight (toWord l) n =
    if length l ≤ width then
      if length l ≤ n then toWord [] else toWord (drop n l)
    else if width ≤ n then toWord []
    else toWord (drop n (take width l))

External Type Operators

• →
• bool
• Data
  • List
    • list
  • Word
    • word
• Number
  • Natural
    • natural

External Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
  • List
    • ::
    • @
    • []
    • drop
    • head
    • length
    • map
    • nth
    • replicate
    • tail
    • take
    • zipWith
  • Word
    • *
    • +
    • <
    • ≤
    • fromNatural
    • modulus
    • toNatural
    • width
• Number
  • Natural
    • *
    • +
    • -
    • <
    • ≤
    • ↑
    • bit0
    • bit1
    • fromBool
    • min
    • mod
    • suc
    • zero
    • Bits
      • Bits.and
      • Bits.bit
      • Bits.bound
      • Bits.compare
      • Bits.cons
      • Bits.fromList
      • Bits.or
      • Bits.shiftLeft
      • Bits.shiftRight
      • Bits.toVector
      • Bits.width

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ Bits.fromList [] = 0

⊦ ∀t. t ⇒ t

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ ∀x. toNatural x < modulus

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀x. replicate x 0 = []

⊦ ∀t. ¬¬t ⇔ t

⊦ ∀t. ⊥ ∧ t ⇔ ⊥

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀x. fromNatural (toNatural x) = x

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 + n = n

⊦ ∀n. n - n = 0

⊦ ∀k. Bits.bound 0 k = 0

⊦ ∀l. l @ [] = l

⊦ modulus = 2 ↑ width

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀x. 0 + x = x

⊦ ∀k. Bits.fromList (replicate ⊥ k) = 0

⊦ ∀l. take (length l) l = l

⊦ ∀l. Bits.width (Bits.fromList l) ≤ length l

⊦ ∀m n. m ≤ m + n

⊦ ∀n k. Bits.bound n k ≤ n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀x. toNatural x mod modulus = toNatural x

⊦ ∀n. toNatural (fromNatural n) = n mod modulus

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀x n. length (replicate x n) = n

⊦ ∀h t. head (h :: t) = h

⊦ ∀h t. tail (h :: t) = t

⊦ ∀n k. length (Bits.toVector n k) = k

⊦ ∀b. fromBool b = if b then 1 else 0

⊦ ∀l. length l = 0 ⇔ l = []

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀m n. min m n = min n m

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀f l. length (map f l) = length l

⊦ ∀l. Bits.fromList l < 2 ↑ length l

⊦ ∀h t. length (h :: t) = suc (length t)

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀n k. Bits.fromList (Bits.toVector n k) = Bits.bound n k

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀n. Bits.shiftLeft n 1 = 2 * n

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀h t. Bits.fromList (h :: t) = Bits.cons h (Bits.fromList t)

⊦ ∀x y. x < y ⇔ toNatural x < toNatural y

⊦ ∀x y. x ≤ y ⇔ toNatural x ≤ toNatural y

⊦ ∀x y. toNatural x = toNatural y ⇒ x = y

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀n k. Bits.toVector (Bits.bound n k) k = Bits.toVector n k

⊦ ∀m n. min m n = if m ≤ n then m else n

⊦ ∀x₁ y₁. fromNatural (x₁ * y₁) = fromNatural x₁ * fromNatural y₁

⊦ ∀x₁ y₁. fromNatural (x₁ + y₁) = fromNatural x₁ + fromNatural y₁

⊦ ∀n k. Bits.bound n k = n ⇔ Bits.width n ≤ k

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀q n. Bits.compare q 0 n ⇔ q ∨ ¬(n = 0)

⊦ ∀n k. Bits.bound n k = n mod 2 ↑ k

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀n l. n ≤ length l ⇒ length (take n l) = n

⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y

⊦ ∀h t. Bits.cons h t = fromBool h + 2 * t

⊦ ∀n k. Bits.width n ≤ k ⇔ n < 2 ↑ k

⊦ ∀t₁ t₂ t₃. (t₁ ∧ t₂) ∧ t₃ ⇔ t₁ ∧ t₂ ∧ t₃

⊦ ∀n j k. Bits.bound (Bits.bound n j) k = Bits.bound n (min j k)

⊦ ∀m n p. m ≤ n ∧ n < p ⇒ m < p

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀l i. Bits.bit (Bits.fromList l) i ⇔ i < length l ∧ nth l i

⊦ ∀l₁ l₂.
    Bits.fromList (l₁ @ l₂) =
    Bits.fromList l₁ + Bits.shiftLeft (Bits.fromList l₂) (length l₁)

⊦ ∀p q r. p ∧ (q ∨ r) ⇔ p ∧ q ∨ p ∧ r

⊦ ∀q m n. Bits.compare q m n ⇔ if q then m ≤ n else m < n

⊦ ∀m n i. Bits.bit (Bits.and m n) i ⇔ Bits.bit m i ∧ Bits.bit n i

⊦ ∀n i k. Bits.bit (Bits.bound n k) i ⇔ i < k ∧ Bits.bit n i

⊦ ∀m n i. Bits.bit (Bits.or m n) i ⇔ Bits.bit m i ∨ Bits.bit n i

⊦ ∀n j k.
    Bits.bound (Bits.shiftLeft n k) (j + k) =
    Bits.shiftLeft (Bits.bound n j) k

⊦ ∀n j k.
    Bits.shiftRight (Bits.bound n (j + k)) k =
    Bits.bound (Bits.shiftRight n k) j

⊦ ∀b f x y. f (if b then x else y) = if b then f x else f y

⊦ ∀l k.
    Bits.shiftRight (Bits.fromList l) k =
    if length l ≤ k then 0 else Bits.fromList (drop k l)

⊦ ∀m n k.
    Bits.toVector (Bits.and m n) k =
    zipWith (∧) (Bits.toVector m k) (Bits.toVector n k)

⊦ ∀m n k.
    Bits.toVector (Bits.or m n) k =
    zipWith (∨) (Bits.toVector m k) (Bits.toVector n k)

⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l

⊦ ∀f l i. i < length l ⇒ nth (map f l) i = f (nth l i)

⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)

⊦ ∀b f. ∃fn. fn [] = b ∧ ∀h t. fn (h :: t) = f h t (fn t)

⊦ ∀l k.
    Bits.toVector (Bits.fromList l) k =
    if length l ≤ k then l @ replicate ⊥ (k - length l) else take k l

⊦ ∀l₁ l₂.
    length l₁ = length l₂ ∧ (∀i. i < length l₁ ⇒ nth l₁ i = nth l₂ i) ⇒
    l₁ = l₂

⊦ ∀q h₁ h₂ t₁ t₂.
    Bits.compare q (Bits.cons h₁ t₁) (Bits.cons h₂ t₂) ⇔
    Bits.compare (¬h₁ ∧ h₂ ∨ ¬(h₁ ∧ ¬h₂) ∧ q) t₁ t₂

OpenTheory package document generated by opentheory 1.3 (release 20150429)