Package char-thm: Properties of Unicode characters

Information

Files

• Package tarball char-thm-1.23.tgz
• Theory source file char-thm.thy (included in the package tarball)

Theorems

⊦ surjective random

⊦ finite universe

⊦ plane = destPlane ∘ dest

⊦ position = destPosition ∘ dest

⊦ ∀c. ∃r. random r = c

⊦ ∀c. ∃n. invariant n ∧ c = mk n

⊦ finite { n. n | invariant n }

⊦ ∀c. plane c < 17

⊦ ∀c₁ c₂. dest c₁ = dest c₂ ⇔ c₁ = c₂

⊦ image mk { n. n | invariant n } = universe

⊦ ∀n. invariant n ⇒ Bits.width n ≤ 21

⊦ ∀c. ∃n. invariant n ∧ c = mk n ∧ dest c = n

⊦ ∀c. dest c = position c + Bits.shiftLeft (plane c) 16

⊦ ∀c. position c < 65534

⊦ size universe = 1111998

⊦ ∀c. dest c < 1114110

⊦ ∀n. destPlane n = 0 ⇔ n < 65536

⊦ ∀n. invariant n ⇒ n < 1114110

⊦ ∀n.
    invariant n ⇔
    let pl ← destPlane n in
    let pos ← destPosition n in
    pos < 65534 ∧
    if ¬(pl = 0) then pl < 17
    else ¬(55296 ≤ pos ∧ pos < 57344) ∧ ¬(64976 ≤ pos ∧ pos < 65008)

External Type Operators

• →
• bool
• Data
  • Char
    • char
  • Pair
    • ×
• Number
  • Natural
    • natural
• Probability
  • Random
    • Probability.Random.random
• Set
  • set

External Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
  • Char
    • dest
    • destPlane
    • destPosition
    • invariant
    • mk
    • plane
    • position
    • random
  • Pair
    • ,
• Function
  • ∘
  • surjective
• Number
  • Natural
    • *
    • +
    • -
    • <
    • ≤
    • ↑
    • bit0
    • bit1
    • div
    • even
    • mod
    • suc
    • zero
    • Bits
      • Bits.bound
      • Bits.shiftLeft
      • Bits.shiftRight
      • Bits.width
    • Uniform
      • Uniform.random
• Set
  • ∅
  • cross
  • \
  • disjoint
  • finite
  • fromPredicate
  • image
  • insert
  • ∩
  • ∈
  • size
  • ⊆
  • ∪
  • universe

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ bit0 0 = 0

⊦ ∀x. x ∈ universe

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. ¬(x ∈ ∅)

⊦ ∀t. t ∨ ¬t

⊦ ∀n. 0 < suc n

⊦ (¬) = λp. 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. t ∨ ⊥ ⇔ t

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀a. mk (dest a) = a

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 * n = 0

⊦ ∀m. m * 0 = 0

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀k. Bits.shiftLeft 0 k = 0

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀c. plane c = destPlane (dest c)

⊦ ∀c. position c = destPosition (dest c)

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. m ↑ 0 = 1

⊦ ∀m. m * 1 = m

⊦ ∀m. 1 * m = m

⊦ ∀m n. m ≤ m + n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀x. size (insert x ∅) = 1

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀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

⊦ ∀m n. ¬(n + m < m)

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀r. invariant r ⇔ dest (mk r) = r

⊦ ∀n. 0 < n ⇔ ¬(n = 0)

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

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀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 - n = m

⊦ ∀n k. Bits.shiftRight (Bits.bound n k) k = 0

⊦ ∀s t. finite s ⇒ finite (s \ t)

⊦ ∀f s. finite s ⇒ finite (image f s)

⊦ ∀n. n ↑ 2 = n * n

⊦ ∀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

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x

⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x

⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y

⊦ ∀t₁ t₂. ¬t₁ ⇒ ¬t₂ ⇔ t₂ ⇒ t₁

⊦ ∀m n. m < n ⇒ m div n = 0

⊦ ∀m n. m < n ⇒ m mod n = m

⊦ ∀n₁ n₂. n₁ ≤ n₂ ⇒ Bits.width n₁ ≤ Bits.width n₂

⊦ ∀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. m + n = m ⇔ n = 0

⊦ ∀k n. Bits.shiftLeft n k = 0 ⇔ n = 0

⊦ ∀s t. disjoint s t ⇔ s ∩ t = ∅

⊦ ∀s t. finite t ∧ s ⊆ t ⇒ finite s

⊦ ∀n. finite { m. m | m < n }

⊦ ∀n. destPlane n = Bits.shiftRight n 16

⊦ ∀n. destPosition n = Bits.bound n 16

⊦ ∀k. Bits.width (2 ↑ k - 1) = k

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀t₁ t₂. ¬(t₁ ∧ t₂) ⇔ ¬t₁ ∨ ¬t₂

⊦ ∀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 * suc n = m + m * n

⊦ ∀m n. m ↑ suc n = m * m ↑ n

⊦ ∀m n. suc m * n = m * n + n

⊦ ∀n k. Bits.shiftRight n k = 0 ⇔ Bits.width n ≤ k

⊦ ∀s t. finite (s ∪ t) ⇔ finite s ∧ finite t

⊦ ∀s t. finite s ∧ finite t ⇒ finite (cross s t)

⊦ ∀p. (∀x. p x) ⇔ ∀a b. p (a, b)

⊦ ∀p. (∃x. p x) ⇔ ∃a b. p (a, b)

⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d

⊦ ∀n i. i < n ⇒ ∃r. Uniform.random n r = i

⊦ ∀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

⊦ ∀n k. Bits.shiftLeft n k = 2 ↑ k * n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀n k. Bits.bound n k + Bits.shiftLeft (Bits.shiftRight n k) k = n

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀m n. m < n ⇔ ∃d. n = m + suc d

⊦ ∀n. size { m. m | m < n } = n

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀n k. Bits.width n ≤ k ⇔ n < 2 ↑ k

⊦ ∀x y z. x = y ∧ y = z ⇒ x = z

⊦ ∀t₁ t₂ t₃. (t₁ ∨ t₂) ∨ t₃ ⇔ t₁ ∨ t₂ ∨ t₃

⊦ ∀m n p. m * (n * p) = n * (m * p)

⊦ ∀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

⊦ ∀p m n. m + p = n + p ⇔ m = n

⊦ ∀m n p. n + m < p + m ⇔ n < p

⊦ ∀m n p. m + n ≤ m + p ⇔ n ≤ p

⊦ ∀m n p. n + m ≤ p + m ⇔ n ≤ p

⊦ ∀k n₁ n₂. Bits.shiftLeft n₁ k ≤ Bits.shiftLeft n₂ k ⇔ n₁ ≤ n₂

⊦ ∀m n p. (m * n + p) mod n = p mod n

⊦ ∀n₁ n₂ k. Bits.bound (n₁ + Bits.shiftLeft n₂ k) k = Bits.bound n₁ k

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀s t. s ⊆ t ⇔ ∀x. x ∈ s ⇒ x ∈ t

⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇔ s = t

⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0

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

⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x

⊦ ∀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

⊦ ∀n₁ n₂ k.
    Bits.shiftRight (n₁ + Bits.shiftLeft n₂ k) k =
    Bits.shiftRight n₁ k + n₂

⊦ ∀s t x. x ∈ s ∩ t ⇔ x ∈ s ∧ x ∈ t

⊦ ∀s t x. x ∈ s ∪ t ⇔ x ∈ s ∨ x ∈ t

⊦ ∀s t x. x ∈ s \ t ⇔ x ∈ s ∧ ¬(x ∈ t)

⊦ ∀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

⊦ ∀s t. finite s ∧ finite t ⇒ size (cross s t) = size s * size t

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

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

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

⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s

⊦ ∀s t. finite s ∧ t ⊆ s ⇒ size (s \ t) = size s - size t

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

⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'

⊦ ∀x y s t. (x, y) ∈ cross s t ⇔ x ∈ s ∧ y ∈ t

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

⊦ ∀s t. finite s ∧ finite t ∧ disjoint s t ⇒ size (s ∪ t) = size s + size t

⊦ ∀f s.
    (∀x y. x ∈ s ∧ y ∈ s ∧ f x = f y ⇒ x = y) ∧ finite s ⇒
    size (image f s) = size s

⊦ ∀n.
    invariant n ⇔
    let pl ← destPlane n in
    let pos ← destPosition n in
    pl < 17 ∧ pos < 65534 ∧
    (pl = 0 ⇒ ¬(55296 ≤ pos ∧ pos < 57344) ∧ ¬(64976 ≤ pos ∧ pos < 65008))

⊦ ∀r.
    random r =
    let n₀ ← Uniform.random 1111998 r in
    let n₁ ← if n₀ < 55296 then n₀ else n₀ + 2048 in
    let n₂ ← if n₁ < 64976 then n₁ else n₁ + 32 in
    let pl ← n₂ div 65534 in
    let pos ← n₂ mod 65534 in
    let n ← pos + Bits.shiftLeft pl 16 in
    mk n

OpenTheory package document generated by opentheory 1.3 (release 20150429)