Package word: Parametric theory of words

Information

nameword
version1.43
descriptionParametric theory of words
authorJoe Hurd <joe@gilith.com>
licenseMIT
requiresbool
natural
list
natural-divides
showData.Bool
Data.List
Data.Word
Data.Word.Bits
Number.Natural

Files

Defined Type Operator

Defined Constants

Theorems

¬(modulus = 0)

w. normal (fromWord w)

w. w w

fromNatural modulus = 0

toWord [] = 0

toNatural (toWord []) = 0

modulus mod modulus = 0

0 mod modulus = 0

w. ¬(w < w)

x. toNatural x < modulus

~0 = 0

x. ~~x = x

x. fromNatural (toNatural x) = x

w. toWord (fromWord w) = w

w. length (fromWord w) = width

n. n mod modulus < modulus

modulus = 2 ^ width

q. compare q [] [] q

x. x + 0 = x

x. x ^ 1 = x

x. 0 + x = x

x. toNatural x div modulus = 0

x. x ^ 0 = 1

x. x * 0 = 0

x. x + ~x = 0

x. 0 * x = 0

x. ~x + x = 0

x. toNatural x mod modulus = toNatural x

x. x * 1 = x

x. 1 * x = x

x. toNatural (fromNatural x) = x mod modulus

l. normal l length l = width

x. ~x = fromNatural (modulus - toNatural x)

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

x y. x * y = y * x

x y. x + y = y + x

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

n. divides modulus n n mod modulus = 0

n. n < modulus n mod modulus = n

x. fromNatural x = 0 divides modulus x

n. n mod modulus mod modulus = n mod modulus

x y. x < y ¬(y x)

x y. x - y = x + ~y

x. ~x = 0 x = 0

l. toNatural (toWord l) < 2 ^ length l

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

x y. x < y toNatural x < toNatural y

x y. x y toNatural x toNatural y

x y. x * ~y = ~(x * y)

x y. ~x * y = ~(x * y)

w1 w2. fromWord w1 = fromWord w2 w1 = w2

x y. ~x = ~y x = y

x y. toNatural x = toNatural y x = y

w1 w2. fromWord w1 = fromWord w2 w1 = w2

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

m n. fromNatural (m ^ n) = fromNatural m ^ n

x y. x + y = x y = 0

x y. y + x = x y = 0

x y. ~x + ~y = ~(x + y)

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

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

x n. x ^ suc n = x * x ^ n

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

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

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

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)

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

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

x y z. x * y * z = x * (y * z)

x y z. x + y + z = x + (y + z)

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

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

w1 w2 w3. w1 < w2 w2 < w3 w1 < w3

w1 w2 w3. w1 < w2 w2 w3 w1 < w3

w1 w2 w3. w1 w2 w2 < w3 w1 < w3

w1 w2 w3. w1 w2 w2 w3 w1 w3

n. 0 ^ n = if n = 0 then 1 else 0

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

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

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

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

x y. fromNatural x = fromNatural y x mod modulus = y mod modulus

x y z. x * (y + z) = x * y + x * z

x y z. (y + z) * x = y * x + z * x

x m n. x ^ m * x ^ n = x ^ (m + n)

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

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)

x y. x < modulus y < modulus fromNatural x = fromNatural y x = y

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

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

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))

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))

Input Type Operators

Input Constants

Assumptions

¬

¬

even 0

odd 0

length [] = 0

bit0 0 = 0

t. t t

n. 0 n

n. n n

p. p

t. t ¬t

n. 0 < suc n

n. n < suc n

n. n suc n

(¬) = λp. p

() = λp. p ((select) p)

t. (x. t) t

t. (λx. t x) = t

() = λp. p = λx.

x. replicate 0 x = []

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

n. ¬(suc n = 0)

n. even n odd n

m. m < 0

n. 0 * n = 0

n. 0 + n = n

m. m + 0 = m

m. m - 0 = m

n. n - n = 0

m. interval m 0 = []

l. [] @ l = l

l. drop 0 l = l

f. map f [] = []

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

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

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

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

h t. head (h :: t) = h

h t. tail (h :: t) = t

n x. length (replicate n x) = n

m n. length (interval m n) = n

n. 0 < n ¬(n = 0)

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

x y. x = y y = x

x y. x = y y = x

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

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

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. 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

n. ¬(n = 0) n mod n = 0

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

t1 t2. ¬t1 ¬t2 t2 t1

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

n. 0 ^ n = if n = 0 then 1 else 0

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. ¬(n = 0) m mod n < n

m n. ¬(n = 0) m div n m

m n. ¬(n = 0) m mod n m

l m. length (l @ m) = length l + length m

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

f g. (x. f x = g x) f = g

() = 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

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

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

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

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

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

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

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

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

a b. ¬(a = 0) (divides a b b mod a = 0)

m n. ¬(n = 0) (m div n = 0 m < n)

m n. ¬(n = 0) m mod n mod n = m mod n

m n. m ^ n = 0 m = 0 ¬(n = 0)

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

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

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

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

(∃!) = λp. () p x y. p x p y x = y

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

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

e f. ∃!fn. fn 0 = e n. fn (suc n) = f (fn n) n

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

P. P [] (a0 a1. P a1 P (a0 :: a1)) x. P x

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

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 i (map f l) = f (nth i l)

m n p. m * n < m * p ¬(m = 0) n < p

NIL' CONS'.
    fn. fn [] = NIL' a0 a1. fn (a0 :: a1) = CONS' a0 a1 (fn a1)

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 i (take n l) = nth i l

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

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

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

l m.
    length l = length m (i. i < length l nth i l = nth i m) l = m

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

k l m.
    k < length l + length m
    nth k (l @ m) = if k < length l then nth k l else nth (k - length l) m

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)