Package word-bits-thm: Properties of word to bit-list conversions

Information

nameword-bits-thm
version1.73
descriptionProperties of word to bit-list conversions
authorJoe Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2012-08-13
requiresbool
list
natural
natural-bits
pair
probability
word-bits-def
word-def
showData.Bool
Data.List
Data.Pair
Data.Word
Data.Word.Bits
Number.Natural
Probability.Random

Files

Theorems

w. normal (fromWord w)

toNatural (toWord []) = 0

w. toWord (fromWord w) = w

w. length (fromWord w) = width

l. toNatural (toWord l) < 2 length l

l. length l = width fromWord (toWord l) = l

w1 w2. fromWord w1 = fromWord w2 w1 = w2

w1 w2. fromWord w1 = fromWord w2 w1 = w2

w1 w2. compare (fromWord w1) (fromWord w2) w1 < w2

w1 w2. compare (fromWord w1) (fromWord w2) w1 w2

l. width length l fromWord (toWord l) = take width l

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

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

w n. bit w n odd (toNatural w div 2 n)

l.
    length l width
    fromWord (toWord l) = l @ replicate (width - length l)

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. 2 * toNatural (toWord l) + 1 < 2 suc (length l)

l n. bit (toWord l) n n < width n < length l nth l n

n.
    fromNatural n =
    toWord
      (if n = 0 then [] else odd n :: fromWord (fromNatural (n div 2)))

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

h t.
    toNatural (toWord (h :: t)) =
    (2 * toNatural (toWord t) + if h then 1 else 0) mod modulus

h t.
    2 * toNatural (toWord t) + (if h then 1 else 0) < 2 suc (length t)

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))

r.
    fromRandom r =
    let (r1, r2) split r in
    let (l, r1') bits width r1 in
    (toWord l, r2)

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))

Input Type Operators

Input Constants

Assumptions

even 0

¬odd 0

¬(modulus = 0)

¬

¬

length [] = 0

bit0 0 = 0

Bits.toNatural [] = 0

t. t t

n. 0 n

n. n n

p. p

toWord [] = 0

t. t ¬t

x. toNatural x < modulus

m. ¬(m < 0)

n. 0 < suc n

n. n < suc n

n. n suc n

(¬) = 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

t. t

x. fromNatural (toNatural x) = x

n. ¬(suc n = 0)

n. even n odd n

n. 0 * n = 0

n. 0 + n = n

m. m + 0 = m

m. m - 0 = m

n. n - n = 0

m. interval m 0 = []

n. n mod modulus n

l. [] @ l = l

l. drop 0 l = l

f. map f [] = []

modulus = 2 width

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

q. compare q [] [] q

x. x + 0 = x

x. toNatural x div modulus = 0

n. even (2 * n)

n. bit1 n = suc (bit0 n)

n. ¬even n odd n

n. ¬odd n even n

m. m 0 = 1

m. m * 1 = m

n. n 1 = n

n. n div 1 = n

n. n mod 1 = 0

m. 1 * m = m

l. take (length l) l = l

m n. m m + n

m n. n m + n

() = λp q. p q p

t. (t ) (t )

n. odd (suc (2 * n))

m. suc m = m + 1

n. even (suc n) ¬even n

m. m 0 m = 0

n. toNatural (fromNatural n) = n mod modulus

l. normal l length l = width

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

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

x y. fst (x, y) = x

x n. length (replicate x n) = n

m n. length (interval m n) = n

p x. p x p ((select) p)

n. 0 < n ¬(n = 0)

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

f y. (let x y in f x) = f y

xy. x y. xy = (x, y)

x y. x = y y = x

h t. nth (h :: t) 0 = h

t1 t2. t1 t2 t2 t1

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 m n

m n. m n n m

m n. m + n - m = n

n r. length (fst (bits n r)) = n

f l. length (map f l) = length l

w. fromWord w = map (bit w) (interval 0 width)

n. n < modulus n mod modulus = n

n. 2 * n = n + n

n. n mod modulus mod modulus = n mod modulus

l. Bits.toNatural l < 2 length l

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. m < suc n m n

m n. suc m n m < n

m. m = 0 n. m = suc n

() = λp q. (λf. f p q) = λf. f

n. even n n mod 2 = 0

n. ¬(n = 0) 0 mod n = 0

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

t1 t2. ¬t1 ¬t2 t2 t1

x y. x < y toNatural x < toNatural y

x y. x y toNatural x toNatural y

x y. toNatural x = toNatural y x = y

w n. bit w n odd (toNatural (shiftRight w n))

m n. m < n m div n = 0

m n. m < n m mod n = m

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

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

m n. m < m + n 0 < n

m n. suc m = suc n m = n

m n. suc m < suc n m < n

m n. suc m suc n m n

m n. m + n = m n = 0

n. odd n n mod 2 = 1

m. Bits.width (2 m - 1) = m

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

m n. even (m * n) even m even n

m n. even (m + n) even m even n

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

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

m n. m suc n = m * m n

m n. ¬(n = 0) m mod n < n

m n. ¬(n = 0) m div n m

l1 l2. length (l1 @ l2) = length l1 + length l2

m n. m n d. n = m + d

() = λp q. r. (p r) (q r) r

m n. m n m < n m = n

m n. n m n + (m - n) = m

m n. n m m - n + n = m

m n. interval m (suc n) = m :: interval (suc m) n

m n. m n n m m = n

n l. n length l length (take n l) = n

f. fn. x y. fn (x, y) = f x y

p. (x y. p x y) y x. p x y

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

x y. toNatural (x + y) = (toNatural x + toNatural y) mod modulus

m n. ¬(m = 0) m * n div m = n

m n. ¬(m = 0) m * n mod m = 0

x y z. x = y y = z x = z

x n i. i < n nth (replicate x n) i = x

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

m n p. m * (n * p) = m * n * p

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

m n p. m + n < m + p n < p

m n p. n + m < p + m n < p

m n p. m + n m + p n p

m n p. m < n n < p m < p

m n p. m < n n p m < p

m n p. m n n < p m < p

m n p. m n n p m p

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

w n. shiftLeft w n = fromNatural (2 n * toNatural w)

w n. shiftRight w n = fromNatural (toNatural w div 2 n)

m n. n < m suc (m - suc n) = m - n

m n. n m (m - n = 0 m = n)

m n. m suc n m = suc n m n

m n. m * n = 0 m = 0 n = 0

f h t. map f (h :: t) = f h :: map f t

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

m n. ¬(n = 0) (m div n = 0 m < n)

m n. m n = 0 m = 0 ¬(n = 0)

m n i. i < n nth (interval m n) i = m + i

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

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

m n. (m mod modulus) * (n mod modulus) mod modulus = m * n mod modulus

m n. (m mod modulus + n mod modulus) mod modulus = (m + n) mod modulus

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

p. p [] (h t. p t p (h :: t)) l. p l

h t n. n < length t nth (h :: t) (suc n) = nth t n

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

f l i. i < length l nth (map f l) i = f (nth l i)

h t. Bits.toNatural (h :: t) = 2 * Bits.toNatural t + if h then 1 else 0

m n p. m * n < m * p ¬(m = 0) n < p

r.
    fromRandom r =
    let (n, r') Uniform.fromRandom modulus r in (fromNatural n, r')

m n p. ¬(n = 0) m * (p mod n) mod n = m * p mod n

m n p. ¬(n * p = 0) m div n div p = m div n * p

m n p. ¬(n * p = 0) m mod n * p mod n = m mod n

n l i. n length l i < n nth (take n l) i = nth l i

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

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

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

l1 l2.
    length l1 = length l2 (i. i < length l1 nth l1 i = nth l2 i)
    l1 = l2

q h1 h2 t1 t2.
    compare q (h1 :: t1) (h2 :: t2)
    compare (¬h1 h2 ¬(h1 ¬h2) q) t1 t2

x m n. x m x n if x = 0 then m = 0 n = 0 else x = 1 m n

n r.
    Uniform.fromRandom n r =
    let w Bits.width (n - 1) in
    let (r1, r2) split r in
    (Uniform.fromRandom.loop n w r1 mod n, r2)

l1 l2 k.
    k < length l1 + length l2
    nth (l1 @ l2) k =
    if k < length l1 then nth l1 k else nth l2 (k - length l1)

n w r.
    Uniform.fromRandom.loop n w r =
    let (l, r') bits w r in
    let m Bits.toNatural l in
    if m < n then m else Uniform.fromRandom.loop n w r'

a b n.
    ¬(n = 0)
    ((a + b) mod n = a mod n + b mod n (a + b) div n = a div n + b div n)