Package list-nth-thm: Properties of the list nth function
Information
| name | list-nth-thm |
| version | 1.57 |
| description | Properties of the list nth function |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2012-12-02 |
| requires | bool list-append list-def list-dest list-last list-length list-map list-nth-def list-set list-thm natural set |
| show | Data.Bool Data.List Number.Natural Set |
Files
- Package tarball list-nth-thm-1.57.tgz
- Theory source file list-nth-thm.thy (included in the package tarball)
Theorems
⊦ ∀l. ¬null l ⇒ nth l (length l - 1) = last l
⊦ ∀p l. all p l ⇔ ∀i. i < length l ⇒ p (nth l i)
⊦ ∀p l. any p l ⇔ ∃i. i < length l ∧ p (nth l i)
⊦ ∀l x. member x l ⇔ ∃i. i < length l ∧ x = nth l i
⊦ ∀l. toSet l = image (nth l) { i. i | i < length l }
⊦ ∀f l i. i < length l ⇒ nth (map f l) i = f (nth l i)
⊦ ∀l1 l2.
length l1 = length l2 ∧ (∀i. i < length l1 ⇒ nth l1 i = nth l2 i) ⇒
l1 = l2
⊦ ∀l1 l2 k.
k < length l1 + length l2 ⇒
nth (l1 @ l2) k =
if k < length l1 then nth l1 k else nth l2 (k - length l1)
External Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- natural
- Natural
- Set
- set
External Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- cond
- ⊥
- ⊤
- List
- ::
- @
- []
- all
- any
- last
- length
- map
- member
- nth
- null
- toSet
- Bool
- Number
- Natural
- +
- -
- <
- ≤
- bit1
- suc
- zero
- Natural
- Set
- ∅
- fromPredicate
- image
- insert
- ∈
Assumptions
⊦ ⊤
⊦ null []
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ length [] = 0
⊦ toSet [] = ∅
⊦ ∀t. t ⇒ t
⊦ ⊥ ⇔ ∀p. p
⊦ fromPredicate (λx. ⊥) = ∅
⊦ ∀m. ¬(m < 0)
⊦ ∀n. ¬(n < n)
⊦ ∀n. 0 < suc n
⊦ ∀n. n < suc n
⊦ (¬) = λp. p ⇒ ⊥
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀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
⊦ ∀n. ¬(suc n = 0)
⊦ ∀m. m - 0 = m
⊦ ∀l. [] @ l = l
⊦ ∀f. image f ∅ = ∅
⊦ ∀l. null l ⇔ l = []
⊦ ∀h t. ¬null (h :: t)
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀n. suc n - 1 = n
⊦ ∀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
⊦ ∀t1 t2. t1 ∧ t2 ⇔ t2 ∧ t1
⊦ ∀f l. length (map f l) = length l
⊦ ∀p x. x ∈ fromPredicate p ⇔ p x
⊦ ∀h t. length (h :: t) = suc (length t)
⊦ ∀m n. ¬(m < n) ⇔ n ≤ m
⊦ ∀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
⊦ ∀h t. toSet (h :: t) = insert h (toSet t)
⊦ ∀m n. suc m + n = suc (m + n)
⊦ ∀m n. suc m = suc n ⇔ m = n
⊦ ∀m n. suc m < suc n ⇔ m < n
⊦ ∀l1 l2. length (l1 @ l2) = length l1 + length l2
⊦ ∀h t. last (h :: t) = if null t then h else last t
⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y
⊦ ∀l h t. (h :: t) @ l = h :: t @ l
⊦ ∀s t. (∀x. x ∈ s ⇔ x ∈ t) ⇔ s = t
⊦ ∀p l. all p l ⇔ ∀x. x ∈ toSet l ⇒ p x
⊦ ∀p l. any p l ⇔ ∃x. x ∈ toSet l ∧ p x
⊦ ∀m n. n ≤ m ⇒ suc m - suc n = m - n
⊦ ∀f h t. map f (h :: t) = f h :: map f t
⊦ ∀f x s. image f (insert x s) = insert (f x) (image f s)
⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n
⊦ ∀p x. x ∈ { y. y | p y } ⇔ p x
⊦ ∀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
⊦ ∀y s f. y ∈ image f s ⇔ ∃x. y = f x ∧ x ∈ s
⊦ ∀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)
⊦ ∀p f s. (∀y. y ∈ image f s ⇒ p y) ⇔ ∀x. x ∈ s ⇒ p (f x)
⊦ ∀p f s. (∃y. y ∈ image f s ∧ p y) ⇔ ∃x. x ∈ s ∧ p (f x)
⊦ ∀n. { m. m | m < suc n } = insert 0 { m. suc m | m < n }