Package natural-prime-sieve-thm: Properties of the sieve of Eratosthenes

Information

Files

• Package tarball natural-prime-sieve-thm-1.17.tgz
• Theory file natural-prime-sieve-thm.thy (included in the package tarball)

Theorems

⊦ max initial = 1

⊦ unfold next initial = Prime.all

⊦ ∀s. primes s = Prime.below (max s + 1)

⊦ ∀s b s'.
    increment s = (b, s') ⇒ max s' = max s + 1 ∧ (b ⇔ prime (max s'))

Input Type Operators

• →
• bool
• Data
  • List
    • list
  • Pair
    • ×
  • Stream
    • stream
• Number
  • Natural
    • natural
    • Prime
      • Sieve
        • sieve

Input Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
  • List
    • ::
    • @
    • []
    • all
    • map
  • Pair
    • ,
    • fst
    • snd
  • Stream
    • ::
    • nth
    • unfold
• Function
  • ∘
• Number
  • Natural
    • *
    • +
    • <
    • ≤
    • bit0
    • bit1
    • div
    • divides
    • even
    • mod
    • prime
    • suc
    • zero
    • Prime
      • Prime.all
      • Prime.below
      • Sieve
        • dest
        • increment
          • increment.inc
        • initial
        • invariant
          • invariant.inv
        • max
        • mk
        • next
        • primes

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ∀p. all p []

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ Prime.below 2 = []

⊦ ∀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. 0 * n = 0

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀a. mk (dest a) = a

⊦ ∀l. [] @ l = l

⊦ ∀l. l @ [] = l

⊦ ∀f. map f [] = []

⊦ ∀n i. invariant.inv n i []

⊦ initial = mk (1, [])

⊦ nth Prime.all 0 = 2

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. 1 * m = m

⊦ ∀s. max s = fst (dest s)

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀m. suc m = m + 1

⊦ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀xy. (fst xy, snd xy) = xy

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀x y. fst (x, y) = x

⊦ ∀x y. snd (x, y) = y

⊦ ∀p x. p x ⇒ p ((select) p)

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

⊦ ∀s. primes s = map fst (snd (dest s))

⊦ ∀xy. ∃x y. xy = (x, y)

⊦ ∀r. invariant r ⇔ dest (mk r) = r

⊦ ∀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 ⇒ m ≤ n

⊦ Prime.below 3 = 2 :: []

⊦ ∀n. 2 * n = n + n

⊦ ∀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

⊦ ∀p. prime p ⇔ ∃i. nth Prime.all i = p

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀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

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m + n = suc (m + n)

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀m n. suc m < suc n ⇔ m < n

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀t₁ t₂. ¬(t₁ ∨ t₂) ⇔ ¬t₁ ∧ ¬t₂

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

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

⊦ ∀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. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀i j. nth Prime.all i < nth Prime.all j ⇔ i < j

⊦ ∀f. ∃fn. ∀x y. fn (x, y) = f x y

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

⊦ ∀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

⊦ ∀x y z. x = y ∧ y = z ⇒ x = z

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

⊦ ∀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 < m + p ⇔ n < p

⊦ ∀n. Prime.below (suc n) = Prime.below n @ if prime n then n :: [] else []

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀s₁ s₂. (∀n. nth s₁ n = nth s₂ n) ⇒ s₁ = s₂

⊦ ∀p f l. all p (map f l) ⇔ all (p ∘ f) l

⊦ ∀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 h t. all p (h :: t) ⇔ p h ∧ all p t

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

⊦ ∀n i. increment.inc n i [] = (⊤, (n, 0, 0) :: [])

⊦ ∀a b. ¬(a = 0) ⇒ (divides a b ⇔ b mod a = 0)

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

⊦ ∀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

⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l

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

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

⊦ ∀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

⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)

⊦ ∀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

⊦ ∀n.
    prime n ⇔ ¬(n = 0) ∧ ¬(n = 1) ∧ all (λp. ¬divides p n) (Prime.below n)

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

⊦ ∀s.
    next s =
    let (b, s') ← increment s in if b then (max s', s') else next s'

⊦ ∀a b n. ¬(n = 0) ⇒ (a mod n + b mod n) mod n = (a + b) mod n

⊦ ∀n a b. ¬(n = 0) ⇒ (suc a mod n = suc b mod n ⇔ a mod n = b mod n)

⊦ ∀n ps.
    invariant (n, ps) ⇔
    ¬(n = 0) ∧ map fst ps = Prime.below (n + 1) ∧ invariant.inv n 0 ps

⊦ ∀n i p k j ps.
    invariant.inv n i ((p, k, j) :: ps) ⇔
    ¬(p = 0) ∧ (k + i) mod p = n mod p ∧ invariant.inv n (i + j) ps

⊦ ∀s.
    increment s =
    let (n, ps) ← dest s in
    let n' ← n + 1 in
    let (b, ps') ← increment.inc n' 1 ps in
    (b, mk (n', ps'))

⊦ ∀n i p k j ps.
    increment.inc n i ((p, k, j) :: ps) =
    let k' ← (k + i) mod p in
    let j' ← j + i in
    if k' = 0 then (⊥, (p, 0, j') :: ps)
    else let (b, ps') ← increment.inc n j' ps in (b, (p, k', 0) :: ps')

OpenTheory package summary generated by opentheory 1.2 (release 20120929)