Package list-replicate: The list replicate function
Information
| name | list-replicate |
| version | 1.56 |
| description | The list replicate function |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| requires | bool list-append list-def list-length list-map list-nth list-set list-thm natural set |
| show | Data.Bool Data.List Number.Natural Set |
Files
- Package tarball list-replicate-1.56.tgz
- Theory source file list-replicate.thy (included in the package tarball)
Defined Constant
- Data
- List
- replicate
- List
Theorems
⊦ ∀x. replicate x 0 = []
⊦ ∀x n. length (replicate x n) = n
⊦ ∀x n. null (replicate x n) ⇔ n = 0
⊦ ∀x n. replicate x (suc n) = x :: replicate x n
⊦ ∀f x n. map f (replicate x n) = replicate (f x) n
⊦ ∀x n i. i < n ⇒ nth (replicate x n) i = x
⊦ ∀x n. toSet (replicate x n) = if n = 0 then ∅ else insert x ∅
⊦ ∀x m n. replicate x (m + n) = replicate x m @ replicate x n
⊦ ∀x n y. member y (replicate x n) ⇔ y = x ∧ ¬(n = 0)
⊦ ∀x n l1 l2.
l1 @ l2 = replicate x n ⇔
replicate x (length l1) = l1 ∧ replicate x (length l2) = l2 ∧
length l1 + length l2 = n
External Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- natural
- Natural
- Set
- set
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- List
- ::
- @
- []
- length
- map
- member
- nth
- null
- toSet
- Bool
- Number
- Natural
- +
- <
- suc
- zero
- Natural
- Set
- ∅
- insert
- ∈
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ length [] = 0
⊦ toSet [] = ∅
⊦ ∀t. t ⇒ t
⊦ ⊥ ⇔ ∀p. p
⊦ ∀x. ¬(x ∈ ∅)
⊦ ∀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 ⇒ ⊤ ⇔ ⊤
⊦ ∀t. ⊥ ∨ t ⇔ t
⊦ ∀t. ⊤ ∨ t ⇔ ⊤
⊦ ∀n. ¬(suc n = 0)
⊦ ∀n. 0 + n = n
⊦ ∀l. [] @ l = l
⊦ ∀f. map f [] = []
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀l. length l = 0 ⇔ null l
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀l. length l = 0 ⇔ l = []
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀x y. x = y ⇒ y = x
⊦ ∀h t. nth (h :: t) 0 = h
⊦ ∀h t. length (h :: t) = suc (length t)
⊦ ∀m. m = 0 ∨ ∃n. m = suc n
⊦ ∀l x. member x l ⇔ x ∈ toSet l
⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀x y. x ∈ insert y ∅ ⇔ x = y
⊦ ∀h t. toSet (h :: t) = insert h (toSet t)
⊦ ∀x s. insert x (insert x s) = insert x s
⊦ ∀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
⊦ ∀t1 t2 t3. (t1 ∧ t2) ∧ t3 ⇔ t1 ∧ t2 ∧ t3
⊦ ∀l h t. (h :: t) @ l = h :: t @ l
⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)
⊦ ∀m n. m + n = 0 ⇔ m = 0 ∧ n = 0
⊦ ∀l1 l2. l1 @ l2 = [] ⇔ l1 = [] ∧ l2 = []
⊦ ∀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
⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l
⊦ ∀h t n. n < length t ⇒ nth (h :: t) (suc n) = nth t n
⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)