Package list-replicate: The list replicate function
Information
| name | list-replicate |
| version | 1.32 |
| description | The list replicate function |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| requires | bool natural set list-length list-set list-nth |
| show | Data.Bool Data.List Number.Natural Set |
Files
- Package tarball list-replicate-1.32.tgz
- Theory file list-replicate.thy (included in the package tarball)
Defined Constant
- Data
- List
- replicate
- List
Theorems
⊦ ∀x. replicate 0 x = []
⊦ ∀n x. length (replicate n x) = n
⊦ ∀x n. replicate (suc n) x = x :: replicate n x
⊦ ∀n x i. i < n ⇒ nth i (replicate n x) = x
⊦ ∀n x. toSet (replicate n x) = if n = 0 then ∅ else insert x ∅
Input Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- natural
- Natural
- Set
- set
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- List
- ::
- []
- length
- nth
- toSet
- Bool
- Number
- Natural
- <
- suc
- zero
- Natural
- Set
- ∅
- insert
Assumptions
⊦ ⊤
⊦ length [] = 0
⊦ toSet [] = ∅
⊦ ⊥ ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∃) = λp. p ((select) p)
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀t. ⊥ ⇒ t ⇔ ⊤
⊦ ∀t. ⊤ ⇒ t ⇔ t
⊦ ∀t. t ⇒ ⊤ ⇔ ⊤
⊦ ∀n. ¬(suc n = 0)
⊦ ∀m. m < 0 ⇔ ⊥
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀h t. nth 0 (h :: t) = 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
⊦ ∀h t. toSet (h :: t) = insert h (toSet t)
⊦ ∀x s. insert x (insert x s) = insert x s
⊦ ∀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 x)
⊦ ∀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 (suc n) (h :: t) = nth n t
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)