Package word: Parametric theory of words

Information

name	word
version	1.1
description	Parametric theory of words
author	Joe Hurd <joe@gilith.com>
license	MIT
show	Data.Bool Data.List Data.Word Number.Numeral

Files

Package tarball word-1.1.tgz
Theory file word.thy (included in the package tarball)

Defined Type Operator

Data
- Word
  - word

Defined Constants

Data
- Word
  - *
  - +
  - -
  - <
  - ≤
  - ¬
  - and
  - bit
  - fromNatural
  - modulus
  - not
  - or
  - shiftLeft
  - shiftRight
  - toNatural
  - Bits
    - Bits.compare
    - Bits.fromWord
    - Bits.normal
    - Bits.toWord

Theorems

⊦ ¬(modulus = 0)

⊦ ∀w. Bits.normal (Bits.fromWord w)

⊦ toNatural (Bits.toWord []) = 0

⊦ ∀x. Number.Natural.< (toNatural x) modulus

⊦ ∀x. fromNatural (toNatural x) = x

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

⊦ ∀w. length (Bits.fromWord w) = width

⊦ ∀n. Number.Natural.< (Number.Natural.mod n modulus) modulus

⊦ modulus = Number.Natural.exp 2 width

⊦ ∀x. Number.Natural.div (toNatural x) modulus = 0

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

⊦ ∀l. Bits.normal l ⇔ length l = width

⊦ ∀x. ¬x = fromNatural (Number.Natural.- modulus (toNatural x))

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

⊦ ∀w. Bits.fromWord w = map (bit w) (interval 0 width)

⊦ ∀n. Number.Natural.< n modulus ⇒ Number.Natural.mod n modulus = n

⊦ ∀n.
Number.Natural.mod (Number.Natural.mod n modulus) modulus =
Number.Natural.mod n modulus

⊦ ∀x y. x < y ⇔ ¬(y ≤ x)

⊦ ∀x y. x - y = x + ¬y

⊦ ∀l.
Number.Natural.< (toNatural (Bits.toWord l))
(Number.Natural.exp 2 (length l))

⊦ ∀l. length l = width ⇔ Bits.fromWord (Bits.toWord l) = l

⊦ ∀x y. x < y ⇔ Number.Natural.< (toNatural x) (toNatural y)

⊦ ∀x y. x ≤ y ⇔ Number.Natural.≤ (toNatural x) (toNatural y)

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

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

⊦ ∀w1 w2. Bits.fromWord w1 = Bits.fromWord w2 ⇒ w1 = w2

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

⊦ ∀w1 w2. Bits.compare F (Bits.fromWord w1) (Bits.fromWord w2) ⇔ w1 < w2

⊦ ∀w1 w2. Bits.compare T (Bits.fromWord w1) (Bits.fromWord w2) ⇔ w1 ≤ w2

⊦ ∀x1 y1.
fromNatural (Number.Natural.* x1 y1) = fromNatural x1 * fromNatural y1

⊦ ∀x1 y1.
fromNatural (Number.Natural.+ x1 y1) = fromNatural x1 + fromNatural y1

⊦ ∀l.
Number.Natural.≤ width (length l) ⇒
Bits.fromWord (Bits.toWord l) = take width l

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

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

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

⊦ ∀x y.
    toNatural (x + y) =
    Number.Natural.mod (Number.Natural.+ (toNatural x) (toNatural y))
      modulus

⊦ ∀n.
    Bits.toWord
      (Number.Natural.odd n ::
       Bits.fromWord (fromNatural (Number.Natural.div n 2))) =
    fromNatural n

⊦ ∀w n.
    bit w n ⇔
    Number.Natural.odd
      (Number.Natural.div (toNatural w) (Number.Natural.exp 2 n))

⊦ ∀w n.
shiftLeft w n =
fromNatural (Number.Natural.* (Number.Natural.exp 2 n) (toNatural w))

⊦ ∀w n.
shiftRight w n =
fromNatural (Number.Natural.div (toNatural w) (Number.Natural.exp 2 n))

⊦ ∀x y.
fromNatural x = fromNatural y ⇔
Number.Natural.mod x modulus = Number.Natural.mod y modulus

⊦ ∀m n.
    Number.Natural.mod
      (Number.Natural.* (Number.Natural.mod m modulus)
         (Number.Natural.mod n modulus)) modulus =
    Number.Natural.mod (Number.Natural.* m n) modulus

⊦ ∀m n.
    Number.Natural.mod
      (Number.Natural.+ (Number.Natural.mod m modulus)
         (Number.Natural.mod n modulus)) modulus =
    Number.Natural.mod (Number.Natural.+ m n) modulus

⊦ ∀l.
    Number.Natural.≤ (length l) width ⇒
    Bits.fromWord (Bits.toWord l) =
    l @ replicate (Number.Natural.- width (length l)) F

⊦ ∀q w1 w2.
Bits.compare q (Bits.fromWord w1) (Bits.fromWord w2) ⇔
if q then w1 ≤ w2 else w1 < w2

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

⊦ ∀l.
    Number.Natural.<
      (Number.Natural.+ (Number.Natural.* 2 (toNatural (Bits.toWord l))) 1)
      (Number.Natural.exp 2 (Number.Natural.suc (length l)))

⊦ ∀x y.
Number.Natural.< x modulus ∧ Number.Natural.< y modulus ∧
fromNatural x = fromNatural y ⇒ x = y

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

⊦ ∀n.
    fromNatural n =
    Bits.toWord
      (if n = 0 then []
       else
         Number.Natural.odd n ::
         Bits.fromWord (fromNatural (Number.Natural.div n 2)))

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

⊦ ∀h t.
    toNatural (Bits.toWord (h :: t)) =
    Number.Natural.mod
      (Number.Natural.+ (Number.Natural.* 2 (toNatural (Bits.toWord t)))
         (if h then 1 else 0)) modulus

⊦ ∀h t.
    Number.Natural.<
      (Number.Natural.+ (Number.Natural.* 2 (toNatural (Bits.toWord t)))
         (if h then 1 else 0))
      (Number.Natural.exp 2 (Number.Natural.suc (length t)))

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

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

⊦ Bits.toWord [] = fromNatural 0 ∧
  ∀h t.
    Bits.toWord (h :: t) =
    if h then shiftLeft (Bits.toWord t) 1 + fromNatural 1
    else shiftLeft (Bits.toWord t) 1

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

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

Input Type Operators

→
bool
Data
- List
  - list
- Pair
  - Data.Pair.*
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - select
  - F
  - T
- List
  - ::
  - @
  - []
  - drop
  - head
  - interval
  - length
  - map
  - nth
  - replicate
  - tail
  - take
  - zipWith
- Pair
  - Data.Pair.,
  - Data.Pair.fst
  - Data.Pair.snd
- Word
  - width
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
  - bit0
  - bit1
  - zero

Assumptions

⊦ T

⊦ ∀n. Number.Natural.≤ 0 n

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

⊦ ∀x. Function.id x = x

⊦ ∀t. t ∨ ¬t

⊦ ∀n. Number.Natural.< 0 (Number.Natural.suc n)

⊦ (¬) = 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

⊦ ∀n. ¬(Number.Natural.suc n = 0)

⊦ ∀n. Number.Natural.even n ∨ Number.Natural.odd n

⊦ ∀m. Number.Natural.+ m 0 = m

⊦ ∀n. Number.Natural.- n n = 0

⊦ ∀n. Number.Natural.even (Number.Natural.* 2 n)

⊦ ∀n. bit0 n = Number.Natural.+ n n

⊦ ∀n. ¬Number.Natural.even n ⇔ Number.Natural.odd n

⊦ ∀n. ¬Number.Natural.odd n ⇔ Number.Natural.even n

⊦ ∀n. Number.Natural.div n 1 = n

⊦ ∀n. Number.Natural.exp n 1 = n

⊦ ∀n. Number.Natural.mod n 1 = 0

⊦ ∀l. take (length l) l = l

⊦ ∀x. (select y. y = x) = x

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

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

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀n. Number.Natural.odd (Number.Natural.suc (Number.Natural.* 2 n))

⊦ ∀m. Number.Natural.suc m = Number.Natural.+ m 1

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

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

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

⊦ ∀h t. tail (h :: t) = t

⊦ ∀n x. length (replicate n x) = n

⊦ ∀m n. length (interval m n) = n

⊦ ∀t h. head (h :: t) = h

⊦ ∀P x. P x ⇒ P ((select) P)

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀n. Number.Natural.< 0 n ⇔ ¬(n = 0)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀x y. x = y ⇒ y = x

⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀a b. (a ⇔ b) ⇒ a ⇒ b

⊦ ∀m n. Number.Natural.* m n = Number.Natural.* n m

⊦ ∀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) = 0

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

⊦ ∀l f. length (map f l) = length l

⊦ ∀n. Number.Natural.* 2 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 = 0 ∨ ∃n. m = Number.Natural.suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀n. Number.Natural.even n ⇔ Number.Natural.mod n 2 = 0

⊦ ∀n. ¬(n = 0) ⇒ Number.Natural.mod 0 n = 0

⊦ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x

⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀m n. Number.Natural.< m n ⇒ Number.Natural.div m n = 0

⊦ ∀m n. Number.Natural.< m n ⇒ Number.Natural.mod m n = m

⊦ ∀m n. Number.Natural.< m (Number.Natural.+ m n) ⇔ Number.Natural.< 0 n

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

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

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

⊦ ∀m n. Number.Natural.+ m n = m ⇔ n = 0

⊦ ∀m n. Number.Natural.- m n = 0 ⇔ Number.Natural.≤ m n

⊦ ∀n. Number.Natural.odd n ⇔ Number.Natural.mod n 2 = 1

⊦ ∀m. Number.Natural.- 0 m = 0 ∧ Number.Natural.- m 0 = m

⊦ ∀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 = 0) ⇒ Number.Natural.< (Number.Natural.mod m n) n

⊦ ∀m n. ¬(n = 0) ⇒ Number.Natural.≤ (Number.Natural.div m n) m

⊦ ∀m n. ¬(n = 0) ⇒ Number.Natural.≤ (Number.Natural.mod m n) m

⊦ ∀l m. length (l @ m) = Number.Natural.+ (length l) (length m)

⊦ ∀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 0 ⇔ T) ∧
∀n. Number.Natural.even (Number.Natural.suc n) ⇔ ¬Number.Natural.even n

⊦ (Number.Natural.odd 0 ⇔ F) ∧
∀n. Number.Natural.odd (Number.Natural.suc n) ⇔ ¬Number.Natural.odd 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

⊦ ∀n l. Number.Natural.≤ n (length l) ⇒ length (take n l) = n

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

⊦ ∀P. (∀x y. P x y) ⇔ ∀y x. P x y

⊦ ∀P Q. P ∧ (∃x. Q x) ⇔ ∃x. P ∧ Q x

⊦ ∀P Q. P ∨ (∃x. Q x) ⇔ ∃x. P ∨ Q x

⊦ ∀m n. ¬(m = 0) ⇒ Number.Natural.div (Number.Natural.* m n) m = n

⊦ ∀m n. ¬(m = 0) ⇒ Number.Natural.mod (Number.Natural.* m n) m = 0

⊦ ∀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

⊦ ∀p q r. p ∧ q ⇒ r ⇔ p ⇒ q ⇒ r

⊦ ∀n x i. Number.Natural.< i n ⇒ nth i (replicate n x) = x

⊦ ∀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 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

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

⊦ ∀P x. (∀y. P y ⇔ y = x) ⇒ (select) P = x

⊦ ∀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

⊦ ∀m n. Number.Natural.* m n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀m n. Number.Natural.+ m n = 0 ⇔ m = 0 ∧ n = 0

⊦ length [] = 0 ∧ ∀h t. length (h :: t) = Number.Natural.suc (length t)

⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀m n. ¬(n = 0) ⇒ (Number.Natural.div m n = 0 ⇔ Number.Natural.< m n)

⊦ ∀m n.
¬(n = 0) ⇒
Number.Natural.mod (Number.Natural.mod m n) n = Number.Natural.mod m n

⊦ ∀m n. Number.Natural.exp m n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀m n i.
Number.Natural.< i n ⇒ nth i (interval m n) = Number.Natural.+ m i

⊦ ∀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 0 = e ∧ ∀n. fn (Number.Natural.suc n) = f (fn n) n

⊦ ∀m n.
    ¬(n = 0) ⇒
    Number.Natural.+ (Number.Natural.* (Number.Natural.div m n) n)
      (Number.Natural.mod m n) = m

⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x

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

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.* m n) (Number.Natural.* m p) ⇔
m = 0 ∨ Number.Natural.≤ n p

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.* m p) (Number.Natural.* n p) ⇔
Number.Natural.≤ m n ∨ p = 0

⊦ ∀f l i. Number.Natural.< i (length l) ⇒ nth i (map f l) = f (nth i l)

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

⊦ (∀x. replicate 0 x = []) ∧
∀n x. replicate (Number.Natural.suc n) x = x :: replicate n x

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

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

⊦ (∀m. interval m 0 = []) ∧
  ∀m n.
    interval m (Number.Natural.suc n) =
    m :: interval (Number.Natural.suc m) n

⊦ (∀m. Number.Natural.exp m 0 = 1) ∧
  ∀m n.
    Number.Natural.exp m (Number.Natural.suc n) =
    Number.Natural.* m (Number.Natural.exp m n)

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

⊦ (∀l. drop 0 l = l) ∧
∀n h t. drop (Number.Natural.suc n) (h :: t) = drop n t

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

⊦ ∀m n p.
    ¬(n = 0) ⇒
    Number.Natural.mod (Number.Natural.* m (Number.Natural.mod p n)) n =
    Number.Natural.mod (Number.Natural.* m p) n

⊦ ∀m n p.
    ¬(Number.Natural.* n p = 0) ⇒
    Number.Natural.div (Number.Natural.div m n) p =
    Number.Natural.div m (Number.Natural.* n p)

⊦ ∀m n p.
    ¬(Number.Natural.* n p = 0) ⇒
    Number.Natural.mod (Number.Natural.mod m (Number.Natural.* n p)) n =
    Number.Natural.mod m n

⊦ ∀n l i.
Number.Natural.≤ n (length l) ∧ Number.Natural.< i n ⇒
nth i (take n l) = nth i l

⊦ (∀m. Number.Natural.< m 0 ⇔ F) ∧
  ∀m n.
    Number.Natural.< m (Number.Natural.suc n) ⇔
    m = n ∨ Number.Natural.< m n

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

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

⊦ (∀f. map f [] = []) ∧ ∀f h t. map f (h :: t) = f h :: map f t

⊦ ∀m n p.
    ¬(n = 0) ⇒
    Number.Natural.mod
      (Number.Natural.* (Number.Natural.mod m n) (Number.Natural.mod p n))
      n = Number.Natural.mod (Number.Natural.* m p) n

⊦ ∀a b n.
    ¬(n = 0) ⇒
    Number.Natural.mod
      (Number.Natural.+ (Number.Natural.mod a n) (Number.Natural.mod b n))
      n = Number.Natural.mod (Number.Natural.+ a b) n

⊦ ∀m n p.
    ¬(Number.Natural.* n p = 0) ⇒
    Number.Natural.mod (Number.Natural.div m n) p =
    Number.Natural.div (Number.Natural.mod m (Number.Natural.* n p)) n

⊦ ∀k l m.
    nth k (l @ m) =
    if Number.Natural.< k (length l) then nth k l
    else nth (Number.Natural.- k (length l)) m

⊦ (∀m. Number.Natural.≤ m 0 ⇔ m = 0) ∧
  ∀m n.
    Number.Natural.≤ m (Number.Natural.suc n) ⇔
    m = Number.Natural.suc n ∨ Number.Natural.≤ m n

⊦ (∀h t. nth 0 (h :: t) = h) ∧
∀h t n. nth (Number.Natural.suc n) (h :: t) = nth n t

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

⊦ ∀l m.
length l = length m ∧
(∀i. Number.Natural.< i (length l) ⇒ nth i l = nth i m) ⇒ l = m

⊦ ∀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

⊦ ∀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 = 0 then m = 0 ⇒ n = 0 else x = 1 ∨ Number.Natural.≤ m n

⊦ ∀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)

⊦ ∀a b n.
    ¬(n = 0) ⇒
    (Number.Natural.mod (Number.Natural.+ a b) n =
     Number.Natural.+ (Number.Natural.mod a n) (Number.Natural.mod b n) ⇔
     Number.Natural.div (Number.Natural.+ a b) n =
     Number.Natural.+ (Number.Natural.div a n) (Number.Natural.div b n))

⊦ (∀n. Number.Natural.+ 0 n = n) ∧ (∀m. Number.Natural.+ m 0 = 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.* 0 n = 0) ∧ (∀m. Number.Natural.* m 0 = 0) ∧
  (∀n. Number.Natural.* 1 n = n) ∧ (∀m. Number.Natural.* m 1 = 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)