Package word16-bytes-thm: word16-bytes-thm

Information

name	word16-bytes-thm
version	1.2
description	word16-bytes-thm
author	Joe Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2011-03-18
show	Data.Bool

Files

Package tarball word16-bytes-thm-1.2.tgz
Theory file word16-bytes-thm.thy (included in the package tarball)

Theorems

⊦ ∀b.
Data.Word16.fromNatural (Data.Byte.toNatural b) =
Data.Word16.Bits.toWord (Data.Byte.Bits.fromWord b)

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

⊦ ∀w. ∃b0 b1. w = Data.Word16.fromBytes b0 b1

⊦ ∀b0 b1.
    Data.Word16.Bits.toWord
      (Data.List.@ (Data.Byte.Bits.fromWord b1)
         (Data.Byte.Bits.fromWord b0)) = Data.Word16.fromBytes b0 b1

⊦ ∀w.
    ∃b0 b1.
      w = Data.Word16.fromBytes b0 b1 ∧
      Data.Word16.toBytes w = Data.Pair., b0 b1

⊦ ∀w.
    Data.Pair.,
      (Data.Byte.Bits.toWord
         (Data.List.drop
            (Number.Numeral.bit0
               (Number.Numeral.bit0
                  (Number.Numeral.bit0
                     (Number.Numeral.bit1 Number.Numeral.zero))))
            (Data.Word16.Bits.fromWord w)))
      (Data.Byte.Bits.toWord
         (Data.List.take
            (Number.Numeral.bit0
               (Number.Numeral.bit0
                  (Number.Numeral.bit0
                     (Number.Numeral.bit1 Number.Numeral.zero))))
            (Data.Word16.Bits.fromWord w))) = Data.Word16.toBytes w

Input Type Operators

→
bool
Data
- Byte
  - Data.Byte.byte
- List
  - Data.List.list
- Pair
  - Data.Pair.*
- Word16
  - Data.Word16.word16
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - select
  - F
  - T
- Byte
  - Data.Byte.bit
  - Data.Byte.fromNatural
  - Data.Byte.modulus
  - Data.Byte.toNatural
  - Data.Byte.width
  - Bits
    - Data.Byte.Bits.fromWord
    - Data.Byte.Bits.toWord
- List
  - Data.List.::
  - Data.List.@
  - Data.List.[]
  - Data.List.drop
  - Data.List.length
  - Data.List.nth
  - Data.List.replicate
  - Data.List.take
  - Data.List.zipWith
- Pair
  - Data.Pair.,
  - Data.Pair.fst
  - Data.Pair.snd
- Word16
  - Data.Word16.+
  - Data.Word16.and
  - Data.Word16.bit
  - Data.Word16.fromBytes
  - Data.Word16.fromNatural
  - Data.Word16.modulus
  - Data.Word16.or
  - Data.Word16.shiftLeft
  - Data.Word16.shiftRight
  - Data.Word16.toBytes
  - Data.Word16.toNatural
  - Data.Word16.width
  - Bits
    - Data.Word16.Bits.fromWord
    - Data.Word16.Bits.toWord
Function
- Function.id
Number
- Natural
  - Number.Natural.*
  - Number.Natural.+
  - Number.Natural.-
  - Number.Natural.<
  - Number.Natural.≤
  - Number.Natural.div
  - Number.Natural.even
  - Number.Natural.exp
  - Number.Natural.mod
  - Number.Natural.odd
  - Number.Natural.suc
- Numeral
  - Number.Numeral.bit0
  - Number.Numeral.bit1
  - Number.Numeral.zero

Assumptions

⊦ T

⊦ ∀x. x = x

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

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

⊦ ∀x. Function.id x = x

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀a. ∃x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

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

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

⊦ ∀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.Natural.- n n = Number.Numeral.zero

⊦ Data.Byte.modulus =
  Number.Natural.exp
    (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero))
    Data.Byte.width

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

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

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

⊦ ∀n.
Number.Natural.mod n (Number.Numeral.bit1 Number.Numeral.zero) =
Number.Numeral.zero

⊦ ∀x. (select y. y = x) = x

⊦ ∀m n. Number.Natural.≤ m (Number.Natural.+ m n)

⊦ ∀m n. Number.Natural.≤ n (Number.Natural.+ m 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

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

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

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

⊦ ∀x.
Data.Byte.toNatural (Data.Byte.fromNatural x) =
Number.Natural.mod x Data.Byte.modulus

⊦ ∀x.
Data.Word16.toNatural (Data.Word16.fromNatural x) =
Number.Natural.mod x Data.Word16.modulus

⊦ ∀x. Data.Pair., (Data.Pair.fst x) (Data.Pair.snd x) = x

⊦ ∀x y. Data.Pair.fst (Data.Pair., x y) = x

⊦ ∀x y. Data.Pair.snd (Data.Pair., x y) = y

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀m n. m = n ⇒ Number.Natural.≤ m n

⊦ ∀m n. Number.Natural.< m n ⇒ Number.Natural.≤ m 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.- m (Number.Natural.+ m n) = Number.Numeral.zero

⊦ ∀m n. Number.Natural.- (Number.Natural.+ m n) m = n

⊦ ∀m n. Number.Natural.- (Number.Natural.+ m n) n = m

⊦ ∀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.< m n ⇔ Number.Natural.≤ n m

⊦ ∀m n. ¬Number.Natural.≤ m n ⇔ Number.Natural.< n m

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

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

⊦ ∀m. m = Number.Numeral.zero ∨ ∃n. m = Number.Natural.suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x

⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀w1 w2. Data.Byte.Bits.fromWord w1 = Data.Byte.Bits.fromWord w2 ⇔ w1 = w2

⊦ ∀w1 w2.
Data.Word16.Bits.fromWord w1 = Data.Word16.Bits.fromWord w2 ⇔ w1 = w2

⊦ ∀m n. Number.Natural.< m n ⇒ Number.Natural.div m n = Number.Numeral.zero

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

⊦ ∀m n. Number.Natural.+ m n = m ⇔ n = Number.Numeral.zero

⊦ ∀m n. Number.Natural.- m n = Number.Numeral.zero ⇔ Number.Natural.≤ m n

⊦ ∀n.
    Number.Natural.odd n ⇔
    Number.Natural.mod n
      (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero)) =
    Number.Numeral.bit1 Number.Numeral.zero

⊦ ∀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

⊦ ∀m n.
¬(n = Number.Numeral.zero) ⇒
Number.Natural.< (Number.Natural.mod m n) n

⊦ ∀P. (∀p. P p) ⇔ ∀p1 p2. P (Data.Pair., p1 p2)

⊦ ∀m n. Number.Natural.≤ m n ⇔ ∃d. n = Number.Natural.+ m d

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

⊦ (∨) = 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

⊦ (Number.Natural.odd Number.Numeral.zero ⇔ F) ∧
∀n. Number.Natural.odd (Number.Natural.suc n) ⇔ ¬Number.Natural.odd n

⊦ ∀w1 w2.
    Data.Word16.and w1 w2 =
    Data.Word16.Bits.toWord
      (Data.List.zipWith (∧) (Data.Word16.Bits.fromWord w1)
         (Data.Word16.Bits.fromWord w2))

⊦ ∀w1 w2.
    Data.Word16.or w1 w2 =
    Data.Word16.Bits.toWord
      (Data.List.zipWith (∨) (Data.Word16.Bits.fromWord w1)
         (Data.Word16.Bits.fromWord w2))

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

⊦ ∀m n.
Number.Natural.odd (Number.Natural.+ m n) ⇔
¬(Number.Natural.odd m ⇔ Number.Natural.odd n)

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

⊦ ∀l n.
Data.Word16.shiftLeft (Data.Word16.Bits.toWord l) n =
Data.Word16.Bits.toWord (Data.List.@ (Data.List.replicate n F) l)

⊦ ∀m n.
Number.Natural.< m n ⇔
∃d. n = Number.Natural.+ m (Number.Natural.suc d)

⊦ ∀P Q. P ∧ (∃x. Q x) ⇔ ∃x. P ∧ Q x

⊦ ∀P Q. P ∨ (∃x. Q x) ⇔ ∃x. P ∨ Q x

⊦ ∀P Q. (∀x. P x ∨ Q) ⇔ (∀x. P x) ∨ Q

⊦ ∀P Q. (∃x. P x) ∧ Q ⇔ ∃x. P x ∧ Q

⊦ ∀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

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

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

⊦ ∀p q r. p ∧ q ⇒ r ⇔ p ⇒ q ⇒ r

⊦ ∀m n p.
Number.Natural.* m (Number.Natural.* n p) =
Number.Natural.* (Number.Natural.* m n) p

⊦ ∀m n p.
Number.Natural.+ m (Number.Natural.+ n p) =
Number.Natural.+ (Number.Natural.+ m n) p

⊦ ∀m n p. Number.Natural.+ m n = Number.Natural.+ m p ⇔ n = p

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.+ m n) (Number.Natural.+ m p) ⇔
Number.Natural.≤ n p

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.+ m p) (Number.Natural.+ n p) ⇔
Number.Natural.≤ m n

⊦ ∀m n p.
Number.Natural.- (Number.Natural.+ m n) (Number.Natural.+ m p) =
Number.Natural.- n p

⊦ ∀m n p.
Number.Natural.< m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.< m p

⊦ ∀m n p.
Number.Natural.≤ m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.≤ m p

⊦ ∀n.
    Data.Word16.Bits.toWord
      (Data.List.:: (Number.Natural.odd n)
         (Data.Word16.Bits.fromWord
            (Data.Word16.fromNatural
               (Number.Natural.div n
                  (Number.Numeral.bit0
                     (Number.Numeral.bit1 Number.Numeral.zero)))))) =
    Data.Word16.fromNatural n

⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀w n.
    Data.Byte.bit w n ⇔
    Number.Natural.odd
      (Number.Natural.div (Data.Byte.toNatural w)
         (Number.Natural.exp
            (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero))
            n))

⊦ ∀w n.
    Data.Word16.bit w n ⇔
    Number.Natural.odd
      (Number.Natural.div (Data.Word16.toNatural w)
         (Number.Natural.exp
            (Number.Numeral.bit0 (Number.Numeral.bit1 Number.Numeral.zero))
            n))

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

⊦ ∀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)

⊦ ∀P.
P Number.Numeral.zero ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀m n.
Number.Natural.exp m n = Number.Numeral.zero ⇔
m = Number.Numeral.zero ∧ ¬(n = Number.Numeral.zero)

⊦ ∀m n p.
Number.Natural.* m (Number.Natural.+ n p) =
Number.Natural.+ (Number.Natural.* m n) (Number.Natural.* m p)

⊦ ∀m n p.
Number.Natural.* m (Number.Natural.- n p) =
Number.Natural.- (Number.Natural.* m n) (Number.Natural.* m p)

⊦ ∀m n p.
Number.Natural.exp m (Number.Natural.+ n p) =
Number.Natural.* (Number.Natural.exp m n) (Number.Natural.exp m p)

⊦ ∀m n p.
Number.Natural.* (Number.Natural.+ m n) p =
Number.Natural.+ (Number.Natural.* m p) (Number.Natural.* n p)

⊦ ∀m n p.
Number.Natural.* (Number.Natural.- m n) p =
Number.Natural.- (Number.Natural.* m p) (Number.Natural.* n p)

⊦ ∀P Q. (∀x. P x ∧ Q x) ⇔ (∀x. P x) ∧ ∀x. Q x

⊦ ∀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 Q. (∃x. P x) ∨ (∃x. Q x) ⇔ ∃x. P x ∨ Q x

⊦ ∀e f.
    ∃fn.
      fn Number.Numeral.zero = e ∧
      ∀n. fn (Number.Natural.suc n) = f (fn n) n

⊦ ∀m n.
    ¬(n = Number.Numeral.zero) ⇒
    Number.Natural.+ (Number.Natural.* (Number.Natural.div m n) n)
      (Number.Natural.mod m n) = m

⊦ ∀w1 w2.
    (∀i.
       Number.Natural.< i Data.Byte.width ⇒
       (Data.Byte.bit w1 i ⇔ Data.Byte.bit w2 i)) ⇒ w1 = w2

⊦ ∀w1 w2.
    (∀i.
       Number.Natural.< i Data.Word16.width ⇒
       (Data.Word16.bit w1 i ⇔ Data.Word16.bit w2 i)) ⇒ w1 = w2

⊦ ∀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

⊦ ∀b1 b2.
    Data.Word16.fromBytes b1 b2 =
    Data.Word16.or
      (Data.Word16.shiftLeft
         (Data.Word16.fromNatural (Data.Byte.toNatural b1))
         (Number.Numeral.bit0
            (Number.Numeral.bit0
               (Number.Numeral.bit0
                  (Number.Numeral.bit1 Number.Numeral.zero)))))
      (Data.Word16.fromNatural (Data.Byte.toNatural b2))

⊦ ∀l n.
    Data.Byte.bit (Data.Byte.Bits.toWord l) n ⇔
    Number.Natural.< n Data.Byte.width ∧
    Number.Natural.< n (Data.List.length l) ∧ Data.List.nth n l

⊦ ∀l n.
    Data.Word16.bit (Data.Word16.Bits.toWord l) n ⇔
    Number.Natural.< n Data.Word16.width ∧
    Number.Natural.< n (Data.List.length l) ∧ Data.List.nth n l

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

⊦ (∀x. Data.List.replicate Number.Numeral.zero x = Data.List.[]) ∧
  ∀n x.
    Data.List.replicate (Number.Natural.suc n) x =
    Data.List.:: x (Data.List.replicate n x)

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

⊦ ∀x y a b. Data.Pair., x y = Data.Pair., a b ⇔ x = a ∧ y = b

⊦ ∀m n p q.
Number.Natural.< m p ∧ Number.Natural.< n q ⇒
Number.Natural.< (Number.Natural.+ m n) (Number.Natural.+ p q)

⊦ ∀m n p q.
Number.Natural.≤ m p ∧ Number.Natural.≤ n q ⇒
Number.Natural.≤ (Number.Natural.+ m n) (Number.Natural.+ p q)

⊦ ∀l.
    Data.Byte.Bits.fromWord (Data.Byte.Bits.toWord l) =
    if Number.Natural.≤ (Data.List.length l) Data.Byte.width then
      Data.List.@ l
        (Data.List.replicate
           (Number.Natural.- Data.Byte.width (Data.List.length l)) F)
    else Data.List.take Data.Byte.width l

⊦ ∀l.
    Data.Word16.Bits.fromWord (Data.Word16.Bits.toWord l) =
    if Number.Natural.≤ (Data.List.length l) Data.Word16.width then
      Data.List.@ l
        (Data.List.replicate
           (Number.Natural.- Data.Word16.width (Data.List.length l)) F)
    else Data.List.take Data.Word16.width l

⊦ (∀m.
     Number.Natural.exp m Number.Numeral.zero =
     Number.Numeral.bit1 Number.Numeral.zero) ∧
  ∀m n.
    Number.Natural.exp m (Number.Natural.suc n) =
    Number.Natural.* m (Number.Natural.exp m n)

⊦ (∀l. Data.List.drop Number.Numeral.zero l = l) ∧
  ∀n h t.
    Data.List.drop (Number.Natural.suc n) (Data.List.:: h t) =
    Data.List.drop n t

⊦ ∀m n p.
    ¬(Number.Natural.* n p = Number.Numeral.zero) ⇒
    Number.Natural.mod (Number.Natural.mod m (Number.Natural.* n p)) n =
    Number.Natural.mod m n

⊦ (∀l. Data.List.@ Data.List.[] l = l) ∧
∀h t l.
Data.List.@ (Data.List.:: h t) l = Data.List.:: h (Data.List.@ t l)

⊦ ∀t1 t2. (¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2) ∧ (¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2)

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

⊦ ∀m n p.
    ¬(Number.Natural.* n p = Number.Numeral.zero) ⇒
    Number.Natural.mod (Number.Natural.div m n) p =
    Number.Natural.div (Number.Natural.mod m (Number.Natural.* n p)) n

⊦ (∀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)

⊦ ∀w.
    Data.Word16.toBytes w =
    Data.Pair.,
      (Data.Byte.fromNatural
         (Data.Word16.toNatural
            (Data.Word16.shiftRight w
               (Number.Numeral.bit0
                  (Number.Numeral.bit0
                     (Number.Numeral.bit0
                        (Number.Numeral.bit1 Number.Numeral.zero)))))))
      (Data.Byte.fromNatural
         (Data.Word16.toNatural
            (Data.Word16.and w
               (Data.Word16.fromNatural
                  (Number.Numeral.bit1
                     (Number.Numeral.bit1
                        (Number.Numeral.bit1
                           (Number.Numeral.bit1
                              (Number.Numeral.bit1
                                 (Number.Numeral.bit1
                                    (Number.Numeral.bit1
                                       (Number.Numeral.bit1
                                          Number.Numeral.zero))))))))))))

⊦ ∀m n q r.
m = Number.Natural.+ (Number.Natural.* q n) r ∧ Number.Natural.< r n ⇒
Number.Natural.div m n = q ∧ Number.Natural.mod m n = r

⊦ Data.Word16.Bits.toWord Data.List.[] =
  Data.Word16.fromNatural Number.Numeral.zero ∧
  ∀h t.
    Data.Word16.Bits.toWord (Data.List.:: h t) =
    if h then
      Data.Word16.+
        (Data.Word16.shiftLeft (Data.Word16.Bits.toWord t)
           (Number.Numeral.bit1 Number.Numeral.zero))
        (Data.Word16.fromNatural (Number.Numeral.bit1 Number.Numeral.zero))
    else
      Data.Word16.shiftLeft (Data.Word16.Bits.toWord t)
        (Number.Numeral.bit1 Number.Numeral.zero)

⊦ ∀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)

⊦ ∀x m n.
    Number.Natural.≤ (Number.Natural.exp x m) (Number.Natural.exp x n) ⇔
    if x = Number.Numeral.zero then
      m = Number.Numeral.zero ⇒ n = Number.Numeral.zero
    else x = Number.Numeral.bit1 Number.Numeral.zero ∨ Number.Natural.≤ m n

⊦ (∀f. Data.List.zipWith f Data.List.[] Data.List.[] = Data.List.[]) ∧
  ∀f h1 h2 t1 t2.
    Data.List.zipWith f (Data.List.:: h1 t1) (Data.List.:: h2 t2) =
    Data.List.:: (f h1 h2) (Data.List.zipWith f t1 t2)

⊦ ∀b.
    ∃x0 x1 x2 x3 x4 x5 x6 x7.
      b =
      Data.Byte.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.[]))))))))

⊦ ∀m n p.
    Number.Natural.* m n = Number.Natural.* n m ∧
    Number.Natural.* (Number.Natural.* m n) p =
    Number.Natural.* m (Number.Natural.* n p) ∧
    Number.Natural.* m (Number.Natural.* n p) =
    Number.Natural.* n (Number.Natural.* m p)

⊦ ∀m n p.
    Number.Natural.+ m n = Number.Natural.+ n m ∧
    Number.Natural.+ (Number.Natural.+ m n) p =
    Number.Natural.+ m (Number.Natural.+ n p) ∧
    Number.Natural.+ m (Number.Natural.+ n p) =
    Number.Natural.+ n (Number.Natural.+ m p)

⊦ ∀l n.
    Data.Word16.shiftRight (Data.Word16.Bits.toWord l) n =
    if Number.Natural.≤ (Data.List.length l) Data.Word16.width then
      if Number.Natural.≤ (Data.List.length l) n then
        Data.Word16.Bits.toWord Data.List.[]
      else Data.Word16.Bits.toWord (Data.List.drop n l)
    else if Number.Natural.≤ Data.Word16.width n then
      Data.Word16.Bits.toWord Data.List.[]
    else
      Data.Word16.Bits.toWord
        (Data.List.drop n (Data.List.take Data.Word16.width l))

⊦ (∀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)

⊦ ∀p q r.
(p ∨ q ⇔ q ∨ p) ∧ ((p ∨ q) ∨ r ⇔ p ∨ q ∨ r) ∧ (p ∨ q ∨ r ⇔ q ∨ p ∨ r) ∧
(p ∨ p ⇔ p) ∧ (p ∨ p ∨ q ⇔ p ∨ q)

⊦ (∀n. Number.Natural.* Number.Numeral.zero n = Number.Numeral.zero) ∧
  (∀m. Number.Natural.* m Number.Numeral.zero = Number.Numeral.zero) ∧
  (∀n. Number.Natural.* (Number.Numeral.bit1 Number.Numeral.zero) n = n) ∧
  (∀m. Number.Natural.* m (Number.Numeral.bit1 Number.Numeral.zero) = m) ∧
  (∀m n.
     Number.Natural.* (Number.Natural.suc m) n =
     Number.Natural.+ (Number.Natural.* m n) n) ∧
  ∀m n.
    Number.Natural.* m (Number.Natural.suc n) =
    Number.Natural.+ m (Number.Natural.* m n)

⊦ ∀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.[]))))))))))))))))