Package option: Basic theory of option types
Information
| name | option |
| version | 1.18 |
| description | Basic theory of option types |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| show | Data.Bool Data.Option Number.Natural |
Files
- Package tarball option-1.18.tgz
- Theory file option.thy (included in the package tarball)
Defined Type Operator
- Data
- Option
- option
- Option
Defined Constants
- Data
- Option
- case
- isNone
- isSome
- none
- some
- Option
Theorems
⊦ ∀a'. ¬(none = some a')
⊦ ∀x. case none some x = x
⊦ ∀x. x = none ∨ ∃a. x = some a
⊦ (isNone none ⇔ T) ∧ ∀a. isNone (some a) ⇔ F
⊦ (isSome none ⇔ F) ∧ ∀a. isSome (some a) ⇔ T
⊦ ∀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
- →
- bool
- Number
- Natural
- natural
- Natural
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- F
- T
- Bool
- Number
- Natural
- *
- +
- <
- ≤
- bit0
- bit1
- even
- exp
- suc
- zero
- Natural
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)