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

Information

name	word16-bytes-thm
version	1.41
description	Properties of 16-bit word to byte pair conversions
author	Joe Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2012-01-29
requires	bool pair natural list byte word16-def word16-bits word16-bytes-def
show	Data.Bool Data.Byte Data.Byte.Bits Data.List Data.Pair Data.Word16 Data.Word16.Bits Number.Natural

Files

Package tarball word16-bytes-thm-1.41.tgz
Theory file word16-bytes-thm.thy (included in the package tarball)

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 b1 @ fromByte b0) = fromBytes b0 b1

⊦ ∀w. ∃b0 b1. w = fromBytes b0 b1 ∧ toBytes w = (b0, b1)

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

Input Type Operators

→
bool
Data
- Byte
  - byte
- List
  - list
- Pair
  - ×
- Word16
  - word16
Number
- Natural
  - natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - F
  - T
- Byte
  - bit
  - fromNatural
  - modulus
  - toNatural
  - width
  - Bits
    - fromByte
    - toByte
- List
  - ::
  - @
  - []
  - drop
  - length
  - nth
  - replicate
  - take
  - zipWith
- Pair
  - ,
  - fst
  - snd
- Word16
  - and
  - bit
  - fromBytes
  - fromNatural
  - modulus
  - or
  - shiftLeft
  - shiftRight
  - toBytes
  - toNatural
  - width
  - Bits
    - fromWord
    - toWord
Number
- Natural
  - *
  - +
  - -
  - <
  - ≤
  - bit0
  - bit1
  - div
  - even
  - exp
  - mod
  - odd
  - suc
  - zero

Assumptions

⊦ T

⊦ ¬F ⇔ T

⊦ ¬T ⇔ F

⊦ odd 0 ⇔ F

⊦ length [] = 0

⊦ bit0 0 = 0

⊦ ∀x. x = x

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ F ⇔ ∀p. p

⊦ toWord [] = 0

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. T

⊦ ∀x. replicate 0 x = []

⊦ ∀t. ¬¬t ⇔ t

⊦ ∀t. (T ⇔ t) ⇔ t

⊦ ∀t. (t ⇔ T) ⇔ t

⊦ ∀t. F ∧ t ⇔ F

⊦ ∀t. T ∧ t ⇔ t

⊦ ∀t. t ∧ F ⇔ F

⊦ ∀t. t ∧ T ⇔ t

⊦ ∀t. t ∧ t ⇔ t

⊦ ∀t. F ⇒ t ⇔ T

⊦ ∀t. T ⇒ t ⇔ t

⊦ ∀t. t ⇒ T ⇔ T

⊦ ∀t. F ∨ t ⇔ t

⊦ ∀t. T ∨ t ⇔ T

⊦ ∀t. t ∨ F ⇔ t

⊦ ∀t. t ∨ T ⇔ T

⊦ ∀w. toByte (fromByte w) = w

⊦ ∀w. length (fromByte w) = width

⊦ ∀w. toWord (fromWord w) = w

⊦ ∀w. length (fromWord w) = width

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 * n = 0

⊦ ∀m. m * 0 = 0

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀l. [] @ l = l

⊦ ∀l. drop 0 l = l

⊦ ∀l. take 0 l = []

⊦ modulus = exp 2 width

⊦ width = 8

⊦ modulus = exp 2 width

⊦ ∀t. (F ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ F) ⇔ ¬t

⊦ ∀t. t ⇒ F ⇔ ¬t

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. exp m 0 = 1

⊦ ∀m. m * 1 = m

⊦ ∀n. n mod 1 = 0

⊦ ∀m. 1 * m = m

⊦ ∀f. zipWith f [] [] = []

⊦ width = 16

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀n. odd (suc n) ⇔ ¬odd n

⊦ ∀m. m ≤ 0 ⇔ m = 0

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

⊦ ∀x. (fst x, snd x) = x

⊦ ∀t1 t2. (if F then t1 else t2) = t2

⊦ ∀t1 t2. (if T then t1 else t2) = t1

⊦ ∀x y. fst (x, y) = x

⊦ ∀x y. snd (x, y) = y

⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀m n. m + n = n + m

⊦ ∀m n. m ≤ n ∨ n ≤ m

⊦ ∀m n. m + n - m = n

⊦ ∀m n. m + n - n = m

⊦ ∀n. 2 * n = n + n

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

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀m n. ¬(m ≤ n ∧ n < m)

⊦ ∀m n. ¬(m < n) ⇔ n ≤ m

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀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 < n ⇒ m div n = 0

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m + n = suc (m + n)

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀n. odd n ⇔ n mod 2 = 1

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

⊦ ∀m n. even (m * n) ⇔ even m ∨ even n

⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n

⊦ ∀m n. m * suc n = m + m * n

⊦ ∀m n. exp m (suc n) = m * exp m n

⊦ ∀m n. suc m * n = m * n + n

⊦ ∀m n. ¬(n = 0) ⇒ m mod n < n

⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d

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

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

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

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. n ≤ m ⇒ n + (m - n) = m

⊦ ∀m n. odd (m + n) ⇔ ¬(odd m ⇔ odd n)

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

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

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀x y z. x = y ∧ y = z ⇒ x = z

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

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

⊦ ∀m n p. m + (n + p) = m + n + p

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀n. toWord (odd n :: fromWord (fromNatural (n div 2))) = fromNatural n

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀w n. bit w n ⇔ odd (toNatural w div exp 2 n)

⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (suc n)) ⇒ ∀n. P n

⊦ ∀m n. exp m n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀m n p. m * (n + p) = m * n + m * p

⊦ ∀m n p. exp m (n + p) = exp m n * exp m p

⊦ ∀m n p. (m + n) * p = m * p + n * p

⊦ ∀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

⊦ ∀m n. ¬(n = 0) ⇒ m div n * n + m mod n = m

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

⊦ ∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t

⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p

⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p

⊦ ∀b1 b2.
    fromBytes b1 b2 =
    or (shiftLeft (fromNatural (toNatural b1)) 8)
      (fromNatural (toNatural b2))

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

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

⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p

⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2

⊦ ∀x y a b. (x, y) = (a, b) ⇔ x = a ∧ y = 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) F
    else take width l

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

⊦ ∀m n p. ¬(n * p = 0) ⇒ m mod (n * p) mod n = m mod n

⊦ ∀m n q r. m = q * n + r ∧ r < n ⇒ m div n = q

⊦ ∀m n q r. m = q * n + r ∧ r < n ⇒ m mod n = r

⊦ ∀m n p. ¬(n * p = 0) ⇒ m div n mod p = m mod (n * p) div n

⊦ ∀w.
    toBytes w =
    (fromNatural (toNatural (shiftRight w 8)),
     fromNatural (toNatural (and w 255)))

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

⊦ ∀x m n.
exp x m ≤ exp x n ⇔ if x = 0 then m = 0 ⇒ n = 0 else x = 1 ∨ m ≤ n

⊦ ∀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 :: [])