Package stream-thm: Properties of infinite stream types

Information

namestream-thm
version1.39
descriptionProperties of infinite stream types
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
provenanceHOL Light theory extracted on 2015-06-30
checksumc51dbedf1b312b3279abe54ba64d13ddeb86365e
requiresbase
stream-def
showData.Bool
Data.List
Data.Pair
Data.Stream
Function
Number.Natural
Set

Files

Theorems

map id = id

s. [] @ s = s

s. drop s 0 = s

a. replicate a = a :: replicate a

s. drop s 1 = tail s

a n. nth (replicate a) n = a

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

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

s n. length (take s n) = n

s. nth s 1 = head (tail s)

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

s. take s 1 = head s :: []

s n. drop s n = (tail n) s

s n. head (drop s n) = nth s n

s1 s2. split (interleave s1 s2) = (s1, s2)

f b. head (unfold f b) = fst (f b)

s n. nth s (suc n) = nth (tail s) n

s n. drop s (suc n) = tail (drop s n)

s n. drop s (suc n) = drop (tail s) n

f a. iterate f a = a :: iterate f (f a)

f g. map f map g = map (f g)

f b. tail (unfold f b) = unfold f (snd (f b))

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

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

l1 l2 s. (l1 @ l2) @ s = l1 @ l2 @ s

s m n. nth s (m + n) = nth (drop s n) m

s1 s2. (n. nth s1 n = nth s2 n) s1 = s2

s1 s2. (n. nth s1 n = nth s2 n) s1 = s2

f g s. map (f g) s = map f (map g s)

s n. take s (suc n) = take s n @ nth s n :: []

s n i. i < n nth (take s n) i = nth s i

s m n. take s (m + n) = take s m @ take (drop s m) n

s n. nth s n = if n = 0 then head s else nth (tail s) (n - 1)

s n x. member x (take s n) i. i < n x = nth s i

s n. take s n = if n = 0 then [] else head s :: take (tail s) (n - 1)

f b. unfold f b = let (a, b') f b in a :: unfold f b'

f b n k. nth (unfold f b) (n + k) = nth (unfold f (((snd f) n) b)) k

l s n.
    nth (l @ s) n = if n < length l then nth l n else nth s (n - length l)

s n. toSet (take s n) = { x. x | i. i < n x = nth s i }

p.
    (m. n. m n p n)
    s. (i j. nth s i nth s j i j) n. p n i. nth s i = n

External Type Operators

External Constants

Assumptions

replicate = iterate id

¬

¬

length [] = 0

bit0 0 = 0

t. t t

n. 0 n

p. p

x. ¬member x []

x. id x = x

t. t ¬t

m. ¬(m < 0)

n. ¬(n < n)

n. 0 < suc n

n. n < suc n

(¬) = λp. p

t. (x. t) t

t. (x. t) t

t. (λx. t x) = t

() = λp. p = λx.

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. 0 + n = n

m. m + 0 = m

m. m - 0 = m

l. [] @ l = l

s. take s 0 = []

s. fromFunction (nth s) = s

f. f 0 = id

f. id f = f

f. nth (fromFunction f) = f

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

n. even (2 * n)

n. bit1 n = suc (bit0 n)

n. ¬odd n even n

s. head s = nth s 0

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. suc n - 1 = n

s. tail s = drop s 1

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

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

x. a b. x = (a, b)

x y. x = y y = x

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

t1 t2. t1 t2 t2 t1

m n. m * n = n * m

m n. m + n = n + m

m n. m + n - m = n

f. iterate f = unfold (λa. (a, f a))

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

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

l x. member x l x toSet l

p. (b. p b) p p

() = 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

t1 t2. ¬t1 ¬t2 t2 t1

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

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

m n. suc m < suc n m < n

f s. map f s = fromFunction (f nth s)

n. odd n n mod 2 = 1

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

t1 t2. ¬(t1 t2) ¬t1 ¬t2

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

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

s n. drop s n = fromFunction (λm. nth s (m + n))

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. m n n m m = n

f. fn. a b. fn (a, b) = f a b

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

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

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

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

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

m n. ¬(m = 0) m * n div m = n

s n. take s (suc n) = head s :: take (tail s) n

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

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

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

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

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

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

m n p. m + n < m + p 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

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

s t. (x. x s x t) s = t

f n x. (f suc n) x = f ((f n) x)

f n x. (f suc n) x = (f n) (f x)

f g h. f (g h) = f g h

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

f b. unfold f b = fromFunction (λn. fst (f (((snd f) n) b)))

p x. x { y. y | p y } p x

x h t. member x (h :: t) x = h member x t

f m n. f (m + n) = f m f n

p. (n. (m. m < n p m) p n) n. p n

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

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

h t. h :: t = fromFunction (λn. if n = 0 then h else nth t (n - 1))

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

p. (n. p n) p ((minimal) p) m. m < (minimal) p ¬p m

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

p c x y. p (if c then x else y) (c p x) (¬c p y)

l s.
    l @ s =
    fromFunction
      (λn. if n < length l then nth l n else nth s (n - length l))

s1 s2.
    interleave s1 s2 =
    fromFunction
      (λn. if even n then nth s1 (n div 2) else nth s2 (n div 2))

s.
    split s =
    (fromFunction (λn. nth s (2 * n)),
     fromFunction (λn. nth s (2 * n + 1)))

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)