Package word16-bits: word16-bits

Information

Files

• Package tarball word16-bits-1.1.tgz
• Theory file word16-bits.thy (included in the package tarball)

Theorem

⊦ ∀w.
    ∃x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15.
      w =
      Data.Word16.Bits.toWord
        (Data.List.:: x0
           (Data.List.:: x1
              (Data.List.:: x2
                 (Data.List.:: x3
                    (Data.List.:: x4
                       (Data.List.:: x5
                          (Data.List.:: x6
                             (Data.List.:: x7
                                (Data.List.:: x8
                                   (Data.List.:: x9
                                      (Data.List.:: x10
                                         (Data.List.:: x11
                                            (Data.List.:: x12
                                               (Data.List.:: x13
                                                  (Data.List.:: x14
                                                     (Data.List.:: x15
                                                        Data.List.[]))))))))))))))))

Input Type Operators

• →
• bool
• Data
  • List
    • Data.List.list
  • Word16
    • Data.Word16.word16
• Number
  • Natural
    • Number.Natural.natural

Input Constants

• =
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • F
    • T
  • List
    • Data.List.::
    • Data.List.[]
    • Data.List.head
    • Data.List.length
    • Data.List.tail
  • Word16
    • Data.Word16.width
    • Bits
      • Data.Word16.Bits.fromWord
      • Data.Word16.Bits.toWord
• Number
  • Natural
    • Number.Natural.*
    • Number.Natural.+
    • Number.Natural.<
    • Number.Natural.≤
    • Number.Natural.even
    • Number.Natural.suc
  • Numeral
    • Number.Numeral.bit0
    • Number.Numeral.bit1
    • Number.Numeral.zero

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ Number.Numeral.zero n

⊦ F ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ F

⊦ ∀t. (∀x. t) ⇔ t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀w. Data.Word16.Bits.toWord (Data.Word16.Bits.fromWord w) = w

⊦ ∀w. Data.List.length (Data.Word16.Bits.fromWord w) = Data.Word16.width

⊦ ∀n. ¬(Number.Natural.suc n = Number.Numeral.zero)

⊦ ∀n. Number.Numeral.bit0 n = Number.Natural.+ n n

⊦ Data.Word16.width =
  Number.Numeral.bit0
    (Number.Numeral.bit0
       (Number.Numeral.bit0
          (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero))))

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀n. Number.Numeral.bit1 n = Number.Natural.suc (Number.Natural.+ n n)

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

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

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀l. Data.List.length l = Number.Numeral.zero ⇔ l = Data.List.[]

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀n.
    Number.Natural.*
      (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero)) n =
    Number.Natural.+ n n

⊦ ∀m n. ¬(Number.Natural.< m n ∧ Number.Natural.≤ n m)

⊦ ∀m n. ¬(Number.Natural.≤ m n ∧ Number.Natural.< n m)

⊦ ∀m n. Number.Natural.≤ (Number.Natural.suc m) n ⇔ Number.Natural.< m n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀m n. Number.Natural.suc m = Number.Natural.suc n ⇔ m = n

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

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

⊦ ∀l. l = Data.List.[] ∨ ∃h t. l = Data.List.:: h t

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

⊦ (Number.Natural.even Number.Numeral.zero ⇔ T) ∧
  ∀n. Number.Natural.even (Number.Natural.suc n) ⇔ ¬Number.Natural.even n

⊦ ∀m n. Number.Natural.≤ m n ⇔ Number.Natural.< m n ∨ m = n

⊦ ∀m n. Number.Natural.≤ m n ∧ Number.Natural.≤ n m ⇔ m = n

⊦ ∀m n.
    Number.Natural.* m n = Number.Numeral.zero ⇔
    m = Number.Numeral.zero ∨ n = Number.Numeral.zero

⊦ Data.List.length Data.List.[] = Number.Numeral.zero ∧
  ∀h t.
    Data.List.length (Data.List.:: h t) =
    Number.Natural.suc (Data.List.length t)

⊦ ∀m n p.
    Number.Natural.* m n = Number.Natural.* m p ⇔
    m = Number.Numeral.zero ∨ n = p

⊦ ∀m n p.
    Number.Natural.≤ (Number.Natural.* m n) (Number.Natural.* m p) ⇔
    m = Number.Numeral.zero ∨ Number.Natural.≤ n p

⊦ ∀m n p.
    Number.Natural.< (Number.Natural.* m n) (Number.Natural.* m p) ⇔
    ¬(m = Number.Numeral.zero) ∧ Number.Natural.< n p

⊦ ∀h1 h2 t1 t2. Data.List.:: h1 t1 = Data.List.:: h2 t2 ⇔ h1 = h2 ∧ t1 = t2

⊦ (∀m. Number.Natural.≤ m Number.Numeral.zero ⇔ m = Number.Numeral.zero) ∧
  ∀m n.
    Number.Natural.≤ m (Number.Natural.suc n) ⇔
    m = Number.Natural.suc n ∨ Number.Natural.≤ m n

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

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

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

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

⊦ (∀n. Number.Natural.+ Number.Numeral.zero n = n) ∧
  (∀m. Number.Natural.+ m Number.Numeral.zero = m) ∧
  (∀m n.
     Number.Natural.+ (Number.Natural.suc m) n =
     Number.Natural.suc (Number.Natural.+ m n)) ∧
  ∀m n.
    Number.Natural.+ m (Number.Natural.suc n) =
    Number.Natural.suc (Number.Natural.+ m n)