Package option-thm: Properties of option types
Information
| name | option-thm |
| version | 1.47 |
| description | Properties of option types |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-11-10 |
| requires | bool natural option-def |
| show | Data.Bool Data.Option Number.Natural |
Files
- Package tarball option-thm-1.47.tgz
- Theory file option-thm.thy (included in the package tarball)
Theorems
⊦ ∀a'. ¬(none = some a')
⊦ ∀x. x = none ∨ ∃a. x = some a
⊦ ∀a a'. some a = some a' ⇔ a = a'
Input Type Operators
- →
- bool
- Data
- Option
- option
- Option
- Number
- Natural
- natural
- Natural
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- ⊥
- ⊤
- Option
- none
- some
- Bool
- Number
- Natural
- *
- +
- <
- ≤
- bit0
- bit1
- even
- suc
- zero
- Natural
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ bit0 0 = 0
⊦ ∀n. 0 ≤ n
⊦ ⊥ ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀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
⊦ ∀n. bit1 n = suc (bit0 n)
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀n. even (suc n) ⇔ ¬even n
⊦ ∀m. m ≤ 0 ⇔ m = 0
⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ ∀n. 2 * n = n + n
⊦ ∀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
⊦ ∀m n. even (m * n) ⇔ even m ∨ even n
⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n
⊦ ∀p. p none ∧ (∀a. p (some a)) ⇒ ∀x. p x
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n
⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n
⊦ ∀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
⊦ ∀b f. ∃fn. fn none = b ∧ ∀a. fn (some a) = f a
⊦ ∀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