Package word16-bytes-thm: Properties of 16-bit word to byte conversions

Information

nameword16-bytes-thm
version1.94
descriptionProperties of 16-bit word to byte conversions
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2015-04-02
checksumf59218a7e6130cc0facec8b896f02f6470870557
requiresbase
byte
natural-bits
word16-bits
word16-bytes-def
word16-def
showData.Bool
Data.Byte
Data.Byte.Bits
Data.List
Data.Pair
Data.Word16
Data.Word16.Bits
Number.Natural

Files

Theorems

b. fromNatural (toNatural b) = toWord (fromByte b)

w. fromNatural (toNatural w) = toByte (fromWord w)

w. b0 b1. w = fromBytes b0 b1

b0 b1. toWord (fromByte b0 @ fromByte b1) = fromBytes b0 b1

w. b0 b1. w = fromBytes b0 b1 toBytes w = (b0, b1)

w.
    (toByte (take 8 (fromWord w)), toByte (drop 8 (fromWord w))) =
    toBytes w

External Type Operators

External Constants

Assumptions

¬

¬

length [] = 0

bit0 0 = 0

x. x = x

t. t t

n. n n

p. p

t. t ¬t

(¬) = λp. p

t. (x. t) t

t. (λx. t x) = t

() = λp. p = λx.

x. replicate x 0 = []

t. ¬¬t t

t. ( t) t

t. (t ) t

t. t

t. t t

t. t t

t. t t t

t. t

t. t t

t. t

t. t t

t. t

t. t t

t. t

n. 0 + n = n

m. m + 0 = m

l. [] @ l = l

l. drop 0 l = l

l. take 0 l = []

width = 8

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

n. bit1 n = suc (bit0 n)

l. toByte l = fromNatural (Bits.fromList l)

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

f. zipWith f [] [] = []

m n. m m + n

width = 16

() = λp q. p q p

t. (t ) (t )

w. fromByte w = Bits.toVector (toNatural w) width

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

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

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

x. (fst x, snd x) = x

t1 t2. (if then t1 else t2) = t2

t1 t2. (if then t1 else t2) = t1

a b. fst (a, b) = a

a b. snd (a, b) = b

n. bit0 (suc n) = suc (suc (bit0 n))

t1 t2. t1 t2 t2 t1

m n. m + n = n + m

m n. m + n - n = m

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

p. ¬(x. p x) x. ¬p x

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

w1 w2. fromByte w1 = fromByte w2 w1 = w2

w1 w2. fromWord w1 = fromWord w2 w1 = w2

m n. m + suc n = suc (m + n)

m n. suc m + n = suc (m + n)

m n. suc m = suc n m = n

x n. replicate x (suc n) = x :: replicate x n

() = λp q. r. (p r) (q r) r

w1 w2. or w1 w2 = toWord (zipWith () (fromWord w1) (fromWord w2))

l n. shiftLeft (toWord l) n = toWord (replicate n @ l)

m n. m < n d. n = m + suc d

p q. p (x. q x) x. p q x

t1 t2 t3. (t1 t2) t3 t1 t2 t3

m n p. m + (n + p) = m + n + p

l h t. (h :: t) @ l = h :: t @ l

p. p 0 (n. p n p (suc n)) n. p n

w.
    toBytes w =
    (fromNatural (toNatural w), fromNatural (toNatural (shiftRight w 8)))

p q. (x. p x q x) (x. p x) x. q x

p q. (x. p x) (x. q x) x. p x q x

n h t. n length t drop (suc n) (h :: t) = drop n t

b0 b1.
    fromBytes b0 b1 =
    or (fromNatural (toNatural b0))
      (shiftLeft (fromNatural (toNatural b1)) 8)

h1 h2 t1 t2. h1 :: t1 = h2 :: t2 h1 = h2 t1 = t2

a b a' b'. (a, b) = (a', b') a = a' b = b'

n h t. n length t take (suc n) (h :: t) = h :: take n t

l.
    fromByte (toByte l) =
    if length l width then l @ replicate (width - length l)
    else take width l

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

f h1 h2 t1 t2.
    length t1 = length t2
    zipWith f (h1 :: t1) (h2 :: t2) = f h1 h2 :: zipWith f t1 t2

b.
    x0 x1 x2 x3 x4 x5 x6 x7.
      b = toByte (x0 :: x1 :: x2 :: x3 :: x4 :: x5 :: x6 :: x7 :: [])

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))

w.
    x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15.
      w =
      toWord
        (x0 :: x1 :: x2 :: x3 :: x4 :: x5 :: x6 :: x7 :: x8 :: x9 :: x10 ::
         x11 :: x12 :: x13 :: x14 :: x15 :: [])