Package gfp-div: GF(p) field division

Information

Files

• Package tarball gfp-div-1.78.tgz
• Theory source file gfp-div.thy (included in the package tarball)

Defined Constants

• Number
  • GF(p)
    • /
    • divGcd
    • expDiv
    • inv

Theorems

⊦ inv 1 = 1

⊦ ∀x. x / 1 = x

⊦ ∀x. ¬(x = 0) ⇒ inv (inv x) = x

⊦ ∀x. ¬(x = 0) ⇒ ¬(inv x = 0)

⊦ ∀x. ¬(x = 0) ⇒ x * inv x = 1

⊦ ∀x. ¬(x = 0) ⇒ inv x * x = 1

⊦ ∀x y. ¬(x = 0) ⇒ x * (y / x) = y

⊦ ∀x y. ¬(x = 0) ⇒ (y / x) * x = y

⊦ ∀x y. ¬(x = 0) ⇒ x * y / x = y

⊦ ∀x y. ¬(x = 0) ⇒ y * x / x = y

⊦ ∀x y. ¬(x = 0) ⇒ y / x = y * inv x

⊦ ∀x n. ¬(x = 0) ⇒ inv x ↑ n = inv (x ↑ n)

⊦ ∀x y. x * y = x ⇔ x = 0 ∨ y = 1

⊦ ∀x y. y * x = x ⇔ x = 0 ∨ y = 1

⊦ ∀x. ¬(x = 0) ∧ inv x = 1 ⇒ x = 1

⊦ ∀x y. ¬(y = 0) ⇒ x / y = divGcd (toNatural y) oddprime x 0

⊦ ∀x y z. x * y = x * z ⇔ x = 0 ∨ y = z

⊦ ∀x y z. y * x = z * x ⇔ x = 0 ∨ y = z

⊦ ∀x y. x * y = if y = 0 then 0 else x / (1 / y)

⊦ ∀x y z. ¬(x = 0) ∧ x * y = x * z ⇒ y = z

⊦ ∀x y z. ¬(x = 0) ∧ y * x = z * x ⇒ y = z

⊦ ∀x y. ¬(x = 0) ∧ ¬(y = 0) ⇒ ¬(y / x = 0)

⊦ ∀x y. ¬(x = 0) ∧ ¬(y = 0) ⇒ inv (y / x) = x / y

⊦ ∀b n d f p. expDiv b n d f p [] = if b then n / d else d / n

⊦ ∀x y. ¬(x = 0) ∧ ¬(y = 0) ∧ inv x = inv y ⇒ x = y

⊦ ∀x y. ¬(x = 0) ∧ ¬(y = 0) ⇒ inv x * inv y = inv (x * y)

⊦ ∀x y z. ¬(y = 0) ∧ ¬(z = 0) ⇒ x / (y / z) = x * z / y

⊦ ∀x n.
    x ↑ n =
    if n = 0 then 1 else if x = 0 then 0 else expDiv ⊤ 1 1 x 1 (encode n)

⊦ ∀b n d f p h t.
    expDiv b n d f p (h :: t) =
    let s ← p / f in expDiv (¬b) d (if h then n / s else n) s f t

⊦ ∀u v x₁ x₂.
    gcd u v = 1 ∧ fromNatural u * x₂ = fromNatural v * x₁ ⇒
    fromNatural u * divGcd u v x₁ x₂ = x₁ ∧
    fromNatural v * divGcd u v x₁ x₂ = x₂

⊦ ∀x n d f p l.
    ¬(x = 0) ∧ ¬(n = 0) ∧ ¬(d = 0) ⇒
    expDiv ⊤ n d (x ↑ f) (inv (x ↑ p)) l =
    (n / d) * x ↑ decode.dest f p l ∧
    expDiv ⊥ n d (inv (x ↑ f)) (x ↑ p) l = (d / n) * x ↑ decode.dest f p l

⊦ ∀u v x₁ x₂.
    divGcd u v x₁ x₂ =
    if u = 1 then x₁
    else if v = 1 then x₂
    else if even u then divGcd (u div 2) v (x₁ / 2) x₂
    else if even v then divGcd u (v div 2) x₁ (x₂ / 2)
    else if v ≤ u then divGcd (u - v) v (x₁ - x₂) x₂
    else divGcd u (v - u) x₁ (x₂ - x₁)

⊦ ∀p.
    (∀v. p 1 v) ∧ (∀u. ¬(u = 1) ⇒ p u 1) ∧
    (∀u v. gcd (2 * u) v = 1 ∧ ¬(v = 1) ∧ p u v ⇒ p (2 * u) v) ∧
    (∀u v. gcd u (2 * v) = 1 ∧ ¬(u = 1) ∧ odd u ∧ p u v ⇒ p u (2 * v)) ∧
    (∀u v. gcd u v = 1 ∧ even u ∧ ¬(v = 1) ∧ odd v ∧ p u v ⇒ p (v + u) v) ∧
    (∀u v. gcd u v = 1 ∧ ¬(u = 1) ∧ odd u ∧ even v ∧ p u v ⇒ p u (u + v)) ⇒
    ∀u v. gcd u v = 1 ⇒ p u v

⊦ ∀p.
    (∀v x₁ x₂. p 1 v x₁ x₂ x₁) ∧ (∀u x₁ x₂. p u 1 x₁ x₂ x₂) ∧
    (∀u v x₁ x₂ g.
       gcd (2 * u) v = 1 ∧ p u v x₁ x₂ g ⇒ p (2 * u) v (2 * x₁) x₂ g) ∧
    (∀u v x₁ x₂ g.
       gcd u (2 * v) = 1 ∧ p u v x₁ x₂ g ⇒ p u (2 * v) x₁ (2 * x₂) g) ∧
    (∀u v x₁ x₂ g.
       gcd u v = 1 ∧ p u v x₁ x₂ g ⇒ p (v + u) v (x₂ + x₁) x₂ g) ∧
    (∀u v x₁ x₂ g.
       gcd u v = 1 ∧ p u v x₁ x₂ g ⇒ p u (u + v) x₁ (x₁ + x₂) g) ⇒
    ∀u v x₁ x₂. gcd u v = 1 ⇒ p u v x₁ x₂ (divGcd u v x₁ x₂)

External Type Operators

• →
• bool
• Data
  • List
    • list
  • Pair
    • ×
• Number
  • GF(p)
    • gfp
  • Natural
    • natural

External Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
  • List
    • ::
    • []
  • Pair
    • ,
• Number
  • GF(p)
    • *
    • +
    • -
    • ↑
    • ~
    • fromNatural
    • oddprime
    • toNatural
  • Natural
    • *
    • +
    • -
    • <
    • ≤
    • bit0
    • bit1
    • div
    • divides
    • even
    • gcd
    • mod
    • odd
    • prime
    • suc
    • zero
    • Fibonacci
      • decode
        • decode.dest
      • encode

Assumptions

⊦ ⊤

⊦ prime oddprime

⊦ ¬(oddprime = 0)

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ ∀t. t ⇒ t

⊦ ⊥ ⇔ ∀p. p

⊦ fromNatural oddprime = 0

⊦ 0 mod oddprime = 0

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀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

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀x. fromNatural (toNatural x) = x

⊦ ∀n. decode (encode n) = n

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀a. gcd 0 a = a

⊦ ∀a. gcd a 0 = a

⊦ ¬(1 = 0)

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀x. x + 0 = x

⊦ ∀x. x ↑ 1 = x

⊦ ∀x. 0 + x = x

⊦ ∀n. even (2 * n)

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀n. ¬even n ⇔ odd n

⊦ ∀n. ¬odd n ⇔ even n

⊦ ∀m. 1 * m = m

⊦ ∀m n. m ≤ m + n

⊦ ¬(2 = 0)

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀x. x ↑ 0 = 1

⊦ ∀x. x * 0 = 0

⊦ ∀x. 0 * x = 0

⊦ ∀x. ~x + x = 0

⊦ ∀x. toNatural x mod oddprime = toNatural x

⊦ ∀x. x * 1 = x

⊦ ∀x. 1 * x = x

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀n. toNatural (fromNatural n) = n mod oddprime

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀n. 0 < n ⇔ ¬(n = 0)

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

⊦ ∀l. decode l = decode.dest 1 0 l

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀t₁ t₂. t₁ ∨ t₂ ⇔ t₂ ∨ t₁

⊦ ∀x y. x * y = y * x

⊦ ∀x y. x + y = y + x

⊦ ∀m n. m + n = n + m

⊦ ∀a b. gcd a b = gcd b a

⊦ ∀m n. m ≤ n ∨ n ≤ m

⊦ ∀m n. m + n - m = n

⊦ ∀x. fromNatural x = 0 ⇔ divides oddprime x

⊦ ∀x y. x - y = x + ~y

⊦ ∀m n. ¬(m ≤ n) ⇔ n < m

⊦ ∀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

⊦ ∀x y. toNatural x = toNatural y ⇒ x = y

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m + n = suc (m + n)

⊦ ∀m n. n < m + n ⇔ 0 < m

⊦ ∀a b. gcd a (a + b) = gcd a b

⊦ ∀a b. gcd (b + a) b = gcd a b

⊦ ∀t₁ t₂. ¬(t₁ ∧ t₂) ⇔ ¬t₁ ∨ ¬t₂

⊦ ∀t₁ t₂. ¬(t₁ ∨ t₂) ⇔ ¬t₁ ∧ ¬t₂

⊦ ∀x n. x ↑ suc n = x * x ↑ 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

⊦ ∀n. even n ⇔ ∃m. n = 2 * m

⊦ ∀p. (∀x. p x) ⇔ ∀a b. p (a, b)

⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀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 q. (∀x. p ⇒ q x) ⇔ p ⇒ ∀x. q x

⊦ ∀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

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀m n. ¬(m = 0) ⇒ m * n div m = n

⊦ ∀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

⊦ ∀p q r. p ⇒ q ⇒ r ⇔ p ∧ q ⇒ r

⊦ ∀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

⊦ ∀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

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

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

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

⊦ ∀p n. prime p ∧ ¬divides p n ⇒ gcd p n = 1

⊦ ∀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)

⊦ ∀p. (∀n. (∀m. m < n ⇒ p m) ⇒ p n) ⇒ ∀n. p n

⊦ ∀p g h. ∃f. ∀x. f x = if p x then f (g x) else h x

⊦ ∀p q. (∃x. p x) ∨ (∃x. q x) ⇔ ∃x. p x ∨ q x

⊦ ∀x y. x * y = 0 ⇔ x = 0 ∨ y = 0

⊦ ∀x n. x ↑ n = 0 ⇔ x = 0 ∧ ¬(n = 0)

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

⊦ ∀m n p. m * p < n * p ⇔ m < n ∧ ¬(p = 0)

⊦ ∀a b. ¬(a = 0) ⇒ ∃s t. t * b + gcd b a = s * a

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

⊦ ∀b f. ∃fn. fn [] = b ∧ ∀h t. fn (h :: t) = f h t (fn t)

⊦ ∀a b c. gcd a (b * c) = 1 ⇔ gcd a b = 1 ∧ gcd a c = 1

⊦ ∀a b c. gcd (b * c) a = 1 ⇔ gcd b a = 1 ∧ gcd c a = 1

⊦ (∀f p. decode.dest f p [] = 0) ∧
  ∀f p h t.
    decode.dest f p (h :: t) =
    let s ← f + p in let n ← decode.dest s f t in if h then s + n else n

OpenTheory package document generated by opentheory 1.3 (release 20150526)