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

Information

name	word16-bytes-thm
version	1.94
description	Properties of 16-bit word to byte conversions
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2015-04-02
checksum	f59218a7e6130cc0facec8b896f02f6470870557
requires	base byte natural-bits word16-bits word16-bytes-def word16-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-1.94.tgz
Theory source 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 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

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

External Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- Byte
  - fromNatural
  - toNatural
  - width
  - Bits
    - fromByte
    - toByte
- List
  - ::
  - @
  - []
  - drop
  - length
  - replicate
  - take
  - zipWith
- Pair
  - ,
  - fst
  - snd
- Word16
  - fromBytes
  - fromNatural
  - or
  - shiftLeft
  - shiftRight
  - toBytes
  - toNatural
  - width
  - Bits
    - fromWord
    - toWord
Number
- Natural
  - +
  - -
  - <
  - ≤
  - bit0
  - bit1
  - suc
  - zero
  - Bits
    - Bits.bound
    - Bits.fromList
    - Bits.toVector

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 :: [])