Package word-bits-def: word-bits-def

Information

Files

• Package tarball word-bits-def-1.0.tgz
• Theory file word-bits-def.thy (included in the package tarball)

Defined Constants

• Data
  • Word
    • Data.Word.and
    • Data.Word.bit
    • Data.Word.not
    • Data.Word.or
    • Data.Word.shiftLeft
    • Data.Word.shiftRight
    • Bits
      • Data.Word.Bits.compare
      • Data.Word.Bits.fromWord
      • Data.Word.Bits.normal
      • Data.Word.Bits.toWord

Theorems

⊦ ∀l. Data.Word.Bits.normal l ⇔ Data.List.length l = Data.Word.width

⊦ ∀w.
    Data.Word.not w =
    Data.Word.Bits.toWord (Data.List.map (¬) (Data.Word.Bits.fromWord w))

⊦ ∀w.
    Data.Word.Bits.fromWord w =
    Data.List.map (Data.Word.bit w)
      (Data.List.interval Number.Numeral.zero Data.Word.width)

⊦ ∀w n.
    Data.Word.bit w n ⇔
    Number.Natural.odd (Data.Word.toNatural (Data.Word.shiftRight w n))

⊦ ∀w1 w2.
    Data.Word.and w1 w2 =
    Data.Word.Bits.toWord
      (Data.List.zipWith (∧) (Data.Word.Bits.fromWord w1)
         (Data.Word.Bits.fromWord w2))

⊦ ∀w1 w2.
    Data.Word.or w1 w2 =
    Data.Word.Bits.toWord
      (Data.List.zipWith (∨) (Data.Word.Bits.fromWord w1)
         (Data.Word.Bits.fromWord w2))

⊦ ∀w n.
    Data.Word.shiftLeft w n =
    Data.Word.fromNatural
      (Number.Natural.*
         (Number.Natural.exp
            (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero))
            n) (Data.Word.toNatural w))

⊦ ∀w n.
    Data.Word.shiftRight w n =
    Data.Word.fromNatural
      (Number.Natural.div (Data.Word.toNatural w)
         (Number.Natural.exp
            (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero))
            n))

⊦ Data.Word.Bits.toWord Data.List.[] =
  Data.Word.fromNatural Number.Numeral.zero ∧
  ∀h t.
    Data.Word.Bits.toWord (Data.List.:: h t) =
    if h then
      Data.Word.+
        (Data.Word.shiftLeft (Data.Word.Bits.toWord t)
           (Number.Numeral.bit1 Number.Numeral.zero))
        (Data.Word.fromNatural (Number.Numeral.bit1 Number.Numeral.zero))
    else
      Data.Word.shiftLeft (Data.Word.Bits.toWord t)
        (Number.Numeral.bit1 Number.Numeral.zero)

⊦ (∀q. Data.Word.Bits.compare q Data.List.[] Data.List.[] ⇔ q) ∧
  ∀q h1 h2 t1 t2.
    Data.Word.Bits.compare q (Data.List.:: h1 t1) (Data.List.:: h2 t2) ⇔
    Data.Word.Bits.compare (¬h1 ∧ h2 ∨ ¬(h1 ∧ ¬h2) ∧ q) t1 t2

Input Type Operators

• →
• bool
• Data
  • List
    • Data.List.list
  • Word
    • Data.Word.word
• Number
  • Natural
    • Number.Natural.natural

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • select
    • F
    • T
  • List
    • Data.List.::
    • Data.List.[]
    • Data.List.head
    • Data.List.interval
    • Data.List.length
    • Data.List.map
    • Data.List.tail
    • Data.List.zipWith
  • Word
    • Data.Word.+
    • Data.Word.fromNatural
    • Data.Word.toNatural
    • Data.Word.width
• Number
  • Natural
    • Number.Natural.*
    • Number.Natural.div
    • Number.Natural.exp
    • Number.Natural.odd
  • Numeral
    • Number.Numeral.bit0
    • Number.Numeral.bit1
    • Number.Numeral.zero

Assumptions

⊦ T

⊦ (∃) = λP. P ((select) P)

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀h t. Data.List.tail (Data.List.:: h t) = t

⊦ ∀t h. Data.List.head (Data.List.:: h t) = h

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀NIL' CONS'.
    ∃fn.
      fn Data.List.[] = NIL' ∧
      ∀a0 a1. fn (Data.List.:: a0 a1) = CONS' a0 a1 (fn a1)

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)