Package option: Option types

Information

nameoption
version1.60
descriptionOption types
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
requiresbool
function
natural
showData.Bool
Data.Option
Function
Number.Natural

Files

Defined Type Operator

Defined Constants

Theorems

isNone none

¬isSome none

map id = id

a. isSome (some a)

a. ¬isNone (some a)

a'. ¬(none = some a')

f. map f none = none

x. case none some x = x

b f. case b f none = b

x. x = none a. x = some a

f a. map f (some a) = some (f a)

a a'. some a = some a' a = a'

b f a. case b f (some a) = f a

f g. map f map g = map (f g)

p. p none (a. p (some a)) x. p x

f g x. map (f g) x = map f (map g x)

b f. fn. fn none = b a. fn (some a) = f a

Input Type Operators

Input Constants

Assumptions

¬

¬

bit0 0 = 0

t. t t

n. 0 n

p. p

x. id x = x

(¬) = λp. p

() = 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. ¬¬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

n. ¬(suc n = 0)

n. 0 + n = n

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

n. even (2 * n)

n. bit1 n = suc (bit0 n)

m. m 0 = 1

() = λp q. p q p

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

n. bit0 (suc n) = suc (suc (bit0 n))

x y. x = y y = x

t1 t2. t1 t2 t2 t1

n. 2 * n = n + n

m n. ¬(m < n n m)

m n. ¬(m n n < m)

m n. suc m n m < n

() = λp q. (λf. f p q) = λf. f

() = λp. q. (x. p x q) q

m n. m + suc n = suc (m + n)

m n. suc m + n = suc (m + n)

m n. suc m = suc n m = n

f g x. (f g) x = f (g x)

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

f g. (x. f x = g x) f = g

p a. (x. a = x p x) p a

() = λp q. r. (p r) (q r) r

m n. m n m < n m = n

m n. m n n m m = n

p q. (x. p q x) p x. q x

t1 t2 t3. (t1 t2) t3 t1 t2 t3

m n p. m * (n * p) = m * n * p

m n p. m + p = n + p m = n

p x. (y. p y y = x) (select) p = x

p. (x. y. p x y) y. x. p x (y x)

m n. m suc n m = suc n m n

m n. m * n = 0 m = 0 n = 0

p. p 0 (n. p n p (suc n)) n. p n

m n. m n = 0 m = 0 ¬(n = 0)

(∃!) = λp. () p x y. p x p y x = y

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 q x) (x. p x) x. q x

p q. (x. p x) (x. q x) x. p x q x

e f. ∃!fn. fn 0 = e n. fn (suc n) = f (fn n) n

m n p. m * n = m * p m = 0 n = p

m n p. m * n m * p m = 0 n p

m n p. m * n < m * p ¬(m = 0) n < p

p1 p2 q1 q2. (p1 p2) (q1 q2) p1 q1 p2 q2

p1 p2 q1 q2. (p1 p2) (q1 q2) p1 q1 p2 q2

p. (x. ∃!y. p x y) f. x y. p x y f x = y

p c x y. p (if c then x else y) (c p x) (¬c p y)

p. (∃!x. p x) (x. p x) x x'. p x p x' x = x'