Package list-map-thm: Properties of the list map function
Information
| name | list-map-thm |
| version | 1.31 |
| description | Properties of the list map function |
| author | Joe Hurd <joe@gilith.com> |
| license | HOLLight |
| provenance | HOL Light theory extracted on 2011-12-18 |
| requires | bool function set list-def list-thm list-length list-set list-append list-map-def |
| show | Data.Bool Data.List Function Number.Natural Set |
Files
- Package tarball list-map-thm-1.31.tgz
- Theory file list-map-thm.thy (included in the package tarball)
Theorems
⊦ map id = id
⊦ ∀l. map (λx. x) l = l
⊦ ∀l f. length (map f l) = length l
⊦ ∀f l. toSet (map f l) = image f (toSet l)
⊦ ∀f l. map f l = [] ⇔ l = []
⊦ ∀f g l. map (g ∘ f) l = map g (map f l)
⊦ ∀f l1 l2. map f (l1 @ l2) = map f l1 @ map f l2
⊦ ∀f. (∀m. ∃l. map f l = m) ⇔ ∀y. ∃x. f x = y
⊦ ∀f. (∀l m. map f l = map f m ⇒ l = m) ⇔ ∀x y. f x = f y ⇒ x = y
Input Type Operators
- →
- bool
- Data
- List
- list
- List
- Number
- Natural
- natural
- Natural
- Set
- set
Input Constants
- =
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- F
- T
- List
- ::
- @
- []
- length
- map
- toSet
- Bool
- Function
- id
- ∘
- Number
- Natural
- suc
- zero
- Natural
- Set
- ∅
- image
- insert
Assumptions
⊦ T
⊦ id = λx. x
⊦ ¬F ⇔ T
⊦ ¬T ⇔ F
⊦ length [] = 0
⊦ toSet [] = ∅
⊦ ∀t. t ⇒ t
⊦ F ⇔ ∀p. p
⊦ ∀x. id x = x
⊦ ∀t. t ∨ ¬t
⊦ (¬) = λp. p ⇒ F
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. T
⊦ ∀t. ¬¬t ⇔ t
⊦ ∀t. (T ⇔ t) ⇔ t
⊦ ∀t. F ∧ t ⇔ F
⊦ ∀t. T ∧ t ⇔ t
⊦ ∀t. t ∧ T ⇔ t
⊦ ∀t. F ⇒ t ⇔ T
⊦ ∀t. T ⇒ t ⇔ t
⊦ ∀t. t ⇒ T ⇔ T
⊦ ∀t. F ∨ t ⇔ t
⊦ ∀t. T ∨ t ⇔ T
⊦ ∀t. t ∨ F ⇔ t
⊦ ∀t. t ∨ T ⇔ T
⊦ ∀l. [] @ l = l
⊦ ∀f. map f [] = []
⊦ ∀f. image f ∅ = ∅
⊦ ∀t. (F ⇔ t) ⇔ ¬t
⊦ ∀t. t ⇒ F ⇔ ¬t
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀h t. ¬(h :: t = [])
⊦ ∀x y. x = y ⇔ y = x
⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1
⊦ ∀h t. length (h :: t) = suc (length t)
⊦ (∧) = λp q. (λf. f p q) = λf. f T T
⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x
⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x
⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q
⊦ ∀h t. toSet (h :: t) = insert h (toSet t)
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀f g. (∀x. f x = g x) ⇔ f = g
⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r
⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x
⊦ ∀p q. (∃x. p x) ∧ q ⇔ ∃x. p x ∧ q
⊦ ∀p q. (∃x. p x) ⇒ q ⇔ ∀x. p x ⇒ q
⊦ ∀t1 t2 t3. (t1 ∨ t2) ∨ t3 ⇔ t1 ∨ t2 ∨ t3
⊦ ∀l h t. (h :: t) @ l = h :: t @ l
⊦ ∀p. (∀x. ∃y. p x y) ⇔ ∃y. ∀x. p x (y x)
⊦ ∀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 [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x
⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2