Package list-map: Definitions and theorems about the list map function
Information
| name | list-map |
| version | 1.12 |
| description | Definitions and theorems about the list map function |
| author | Joe Hurd <joe@gilith.com> |
| license | MIT |
| show | Data.Bool Data.List Function |
Files
- Package tarball list-map-1.12.tgz
- Theory file list-map.thy (included in the package tarball)
Defined Constant
- Data
- List
- map
- List
Theorems
⊦ map id = id
⊦ ∀l. map (λx. x) l = l
⊦ ∀l f. length (map f l) = length l
⊦ ∀f l. toSet (map f l) = Set.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. map f [] = []) ∧ ∀f h t. map f (h :: t) = f h :: map f t
⊦ ∀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
- Number.Natural.natural
- Natural
- Set
- Set.set
Input Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∨
- ¬
- F
- T
- List
- ::
- @
- []
- length
- toSet
- Bool
- Function
- id
- ∘
- Number
- Natural
- Number.Natural.suc
- Number.Natural.zero
- Natural
- Set
- Set.∅
- Set.image
- Set.insert
Assumptions
⊦ T
⊦ id = λx. x
⊦ F ⇔ ∀p. p
⊦ ∀x. id x = x
⊦ ∀t. t ∨ ¬t
⊦ (¬) = λp. p ⇒ F
⊦ (∃) = λP. P ((select) P)
⊦ ∀t. (∀x. t) ⇔ t
⊦ ∀t. (λx. t x) = t
⊦ (∀) = λp. p = λx. T
⊦ ∀x. x = x ⇔ T
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)
⊦ ∀h t. ¬(h :: t = [])
⊦ (¬T ⇔ F) ∧ (¬F ⇔ 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
⊦ ∀f g x. (f ∘ g) x = f (g x)
⊦ ∀f g. f = g ⇔ ∀x. f x = g x
⊦ (∨) = 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
⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)
⊦ length [] = 0 ∧ ∀h t. length (h :: t) = Number.Natural.suc (length t)
⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)
⊦ toSet [] = Set.∅ ∧ ∀h t. toSet (h :: t) = Set.insert h (toSet t)
⊦ ∀P. P [] ∧ (∀a0 a1. P a1 ⇒ P (a0 :: a1)) ⇒ ∀x. P x
⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2
⊦ ∀NIL' CONS'.
∃fn. fn [] = NIL' ∧ ∀a0 a1. fn (a0 :: a1) = CONS' a0 a1 (fn a1)
⊦ (∀l. [] @ l = l) ∧ ∀l h t. (h :: t) @ l = h :: t @ l
⊦ (∀f. Set.image f Set.∅ = Set.∅) ∧
∀f x s. Set.image f (Set.insert x s) = Set.insert (f x) (Set.image f s)
⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)
⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)
⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)
⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)
⊦ ∀p q r.
(p ∨ q ⇔ q ∨ p) ∧ ((p ∨ q) ∨ r ⇔ p ∨ q ∨ r) ∧ (p ∨ q ∨ r ⇔ q ∨ p ∨ r) ∧
(p ∨ p ⇔ p) ∧ (p ∨ p ∨ q ⇔ p ∨ q)