Package list-replicate-thm: Properties of the list replicate function
Information
| name | list-replicate-thm |
| version | 1.41 |
| description | Properties of the list replicate function |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-06-08 |
| requires | bool list-length list-nth list-replicate-def list-set natural set |
| show | Data.Bool Data.List Number.Natural Set |
Files
- Package tarball list-replicate-thm-1.41.tgz
- Theory file list-replicate-thm.thy (included in the package tarball)
Theorems
⊦ ∀x n. length (replicate x n) = 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 n y. member y (replicate x n) ⇔ y = x ∧ ¬(n = 0)
Input Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- natural
- Natural
- Set
- set
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- cond
- ⊥
- ⊤
- List
- ::
- []
- length
- member
- nth
- replicate
- toSet
- Bool
- Number
- Natural
- <
- suc
- zero
- Natural
- Set
- ∅
- insert
- ∈
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ length [] = 0
⊦ toSet [] = ∅
⊦ ∀t. t ⇒ t
⊦ ⊥ ⇔ ∀p. p
⊦ ∀x. ¬(x ∈ ∅)
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀t. (∀x. t) ⇔ t
⊦ (∀) = λp. p = λx. ⊤
⊦ ∀x. replicate x 0 = []
⊦ ∀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)
⊦ ∀m. m < 0 ⇔ ⊥
⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ ⊥ ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀t1 t2. (if ⊥ then t1 else t2) = t2
⊦ ∀t1 t2. (if ⊤ then t1 else t2) = t1
⊦ ∀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 < suc n ⇔ m < n
⊦ ∀x n. replicate x (suc n) = x :: replicate x n
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ ∀h t n. n < length t ⇒ nth (h :: t) (suc n) = nth t n
⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)