Package list-interval: The list interval function
Information
| name | list-interval |
| version | 1.54 |
| description | The list interval function |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| requires | bool list-length list-map list-nth natural |
| show | Data.Bool Data.List Number.Natural |
Files
- Package tarball list-interval-1.54.tgz
- Theory source file list-interval.thy (included in the package tarball)
Defined Constant
- Data
- List
- interval
- List
Theorems
⊦ ∀m. interval m 0 = []
⊦ ∀m n. length (interval m n) = n
⊦ ∀m n. map suc (interval m n) = interval (suc m) n
⊦ ∀m n. interval m (suc n) = m :: interval (suc m) n
⊦ ∀m n i. i < n ⇒ nth (interval m n) i = m + i
External Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- natural
- Natural
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- ⊥
- ⊤
- List
- ::
- []
- length
- map
- nth
- Bool
- Number
- Natural
- +
- <
- suc
- zero
- Natural
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ length [] = 0
⊦ ∀t. t ⇒ t
⊦ ⊥ ⇔ ∀p. p
⊦ ∀m. ¬(m < 0)
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∃) = λp. p ((select) 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 ⇔ ⊤
⊦ ∀m. m + 0 = m
⊦ ∀f. map f [] = []
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀h t. nth (h :: t) 0 = h
⊦ ∀h t. length (h :: t) = suc (length t)
⊦ ∀m. m = 0 ∨ ∃n. m = suc 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. suc m < suc n ⇔ m < n
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y
⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)
⊦ ∀f h t. map f (h :: t) = f h :: map f t
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y
⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n
⊦ ∀h t n. n < length t ⇒ nth (h :: t) (suc n) = nth t n