Package option: Basic theory of option types

Information

nameoption
version1.0
descriptionBasic theory of option types
authorJoe Hurd <joe@gilith.com>
licenseMIT
show Data.Bool
Data.Option
Number.Natural
Number.Numeral

Files

Defined Type Operator

Defined Constants

Theorems

a'. ¬(none = some a')

x. x = none a. x = some a

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

P. P none (a. P (some a)) x. P x

NONE' SOME'. fn. fn none = NONE' a. fn (some a) = SOME' a

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

Input Type Operators

Input Constants

Assumptions

T

n. 0 n

F p. p

(¬) = 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. ¬(suc n = 0)

n. even (2 * n)

n. bit0 n = n + n

() = λp q. p q p

t. (t T) (t F)

n. bit1 n = suc (n + n)

(¬T F) (¬F T)

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 T T

() = λP. q. (x. P x q) q

m n. suc m = suc n m = n

m n. even (m * n) even m even n

m n. even (m + n) even m even n

f g. f = g x. f x = g x

P a. (x. a = x P x) P a

() = λp q. r. (p r) (q r) r

(even 0 T) n. even (suc n) ¬even n

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)

t1 t2. (if T then t1 else t2) = t1 (if F then t1 else t2) = t2

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

P. P 0 (n. P n P (suc n)) n. P n

(t. ¬¬t t) (¬T F) (¬F T)

m n. exp m n = 0 m = 0 ¬(n = 0)

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

A B C D. (A B) (C D) A C B D

A B C D. (A B) (C D) A C B D

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

(m. exp m 0 = 1) m n. exp m (suc n) = m * exp m n

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'

(m. m 0 m = 0) m n. m suc n m = suc n 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. 0 + n = n) (m. m + 0 = m) (m n. suc m + n = suc (m + n))
  m n. m + suc n = suc (m + n)